Attitude Dynamics and Stability of a Simple Solar Photon Thruster

Abstract

This paper is dedicated to the development of a model of the attitude dynamics for a nonideal Simple Solar Photon Thruster (SSPT) and to the analysis of sailcraft motions with respect to their centre of mass. Derivation of the expressions for force and torque due to solar radiation that is valid for the case, when there is a misalignment of the SSPT axis with the sun direction, is followed by study of sailcraft dynamics and stability properties. Analysis of stability shows that an ideally reflecting sail is unstable, while for a sailcraft with nonideal collector, the symmetry axis is stable with respect to the Sun direction for large variety of system parameters. The motion around symmetry axis is always unstable and requires an active stabilizer.

1. Introduction

Solar sails are attractive spacecrafts for long missions, for they create a propulsion effort using freely available solar radiation pressure. The initial concepts of such vehicle have been proposed in the 1920s [1, 2] as an alternative for minimizing on-board energy consumption. With the recent development of new films, materials, and structures its implementation has become possible making solar sails a subject of detailed research regarding eventual missions, the respective control issues, the sailcraft structure, lab testing, and so forth (see, e.g., [3–12], just to name a few), and leading to the implementation of the first sailcraft mission by JAXA in 2010 [10].

The large majority of these studies are focused on the analysis of the Flat Solar Sail (FSS); its principal element is a large flat surface formed by a reflecting membrane. Control of FSS orbital motion is performed by turning the entire sail with respect to the sunlight direction. As a result, the radiation pressure on the surface changes, enabling the control of the vehicle trajectory. Attitude dynamics of such sailcraft are studied, for example, in [4, 11, 12].

Another possible scheme for a sailcraft is the Solar Photon Thruster (SPT). Its main idea is to separate the functions of collecting and reflecting the incoming light. Even though the SSPT's general scheme has been proposed long ago [1, 2, 13], the detailed research has begun quite recently [14–22]. SPT consists of a parabolic reflecting surface (collector) that is always pointing towards the sun, taking advantage of the totality of the sail area. The incoming light is then reflected on a system of smaller mirrors capable of controling the direction of the outgoing light; the sailcraft is, therefore, controlled more easily by the rotation of one smaller mirror.

The SPT can be either Simple SPT (SSPT) or Dual Reflector SPT (DR SPT) depending on the system of smaller mirrors used. The SSPT is provided with two main elements, the Collector C and the Director mirror D [1, 13–15, 18, 19]. Ideally, the symmetry axis of the collector C remains aligned to the sun-sailcraft direction, and all incoming light is concentrated on the collector focus. Director D is placed at this focus; it can rotate changing the direction of the outgoing light in order to control the force acting on the sailcraft and therefore its motion.

Several versions of DR SPT structure (see, e.g., [1, 13, 19–21]) include one more parabolic mirror, called Reflector R. The incoming light from C hits R, becoming a parallel beam that is reflected on D. The director can rotate controlling the resulting propulsion.

Previous studies [17, 19] show that SSPT has many advantages compared to DR SPT. Although its heat tolerance requirements are higher than DR SPT (all the incoming light is concentrated at one point of the director), the advantages are more numerous. When compared to DR SPT, SSPT is much simpler, being therefore easier to deploy and maintain during the flight, and more compact and lighter, thus saving space for the payload. SSPT is more robust than DR SPT regarding the deformations of the collector shape and misalignments with respect to sun-sailcraft direction. For the above reasons, SSPT has been chosen to be the subject of more detailed studies.

In [22], the first model of force acting on SSPT which is assumed to be perfectly aligned with the sun-sailcraft direction is presented and some trajectory simulations are performed. Here, elaboration of a force and torque model for SSPT attending to the eventual axis misalignment permits one to study the sailcraft attitude dynamics and stability properties.

2. Force and Torque Model for a Nonideal Sailcraft

Figure 1 shows the general scheme of SSPT. The Collector C is a big parabolic reflecting surface which concentrates the parallel beam of the incoming sunlight at its focus 𝐹. Director D is a relatively small plane mirror located at the focus of the collector. The force acting on the sail is controlled by rotating D which changes the direction of outgoing light varying the magnitude and direction of the total propulsion force.

Figure 1 

Scheme of a Simple Solar Photon Thruster.

In [22], the general characteristics of a compound solar sail have been analysed. The results show that for minimizing collector film overheating in order to lower the thermal requirements, the light beam should not reflect on the collector more than once. To avoid secondary reflections, the angle 𝛼 between the collector’s symmetry axis and the normal to the director’s surface must satisfy the following condition: tan𝛼>4𝑓_𝑐𝑅_𝑐4𝑓²_𝑐−𝑅²_𝑐. (1) Here, 𝑓_𝑐 and 𝑅_𝑐 are the focal distance and the effective radius of the collector, respectively. If the control angle is limited by condition (1), the components of the force and torque due to solar radiation pressure can be obtained analytically.

A force and torque model for SSPT has been developed in [22] assuming that all sailcraft elements possess ideally reflecting surfaces. But as the collector is a large surface, it should be made of a thin film and it cannot be an ideal reflector. The director is much smaller, so it can have better optical properties. So, here a model for force and torque considering a nonideal collector is developed. The director is supposed to be ideal and small enough so that the effect of its shadow can be disregarded. Consider a SSPT whose symmetry axis is misaligned in relation to sunlight direction.

To calculate the vectors of force and torque due to solar radiation on the SSPT, consider a parabolic surface (collector) shown in Figure 2. Its equation in the reference frame 𝑂𝑥𝑦𝑧, in which the center of mass 𝑂 coincides with the focus 𝐹 of the paraboloid and 𝑂𝑧 is its symmetry axis, is given by 𝑥²+𝑦²+4𝑓_𝑐𝑧−𝑓_𝑐=0, (2) or 1𝑧=4𝑓_𝑐4𝑓_𝑐−𝑥²−𝑦²=0. (3)

Figure 2 

Scheme of reflections on SSPT with axis misalignment.

Suppose that the incoming sunlight has the direction 𝜎 (Figure 2). It is absorbed by the collector at the point 𝐴_𝑐 of the element of the parabolic surface 𝑑𝑆 with the position vector 𝜉=𝑥𝑖⃗+𝑦𝑗+𝑧𝑘, producing the force 𝑑𝑃₁=−𝜌𝜎(𝜎⋅𝑛)𝑑𝑆, (4) where 𝜌=Φ𝑟²_𝐸/𝑟²_𝑠 is the intensity of the light flow at the current point of the orbit. The respective torque about the center of the mass of the SSPT 𝑂 is 𝑑𝑀₁=𝜉×𝑑𝑃₁. (5)

A fraction 𝑟̃ of the incident light is reflected on the element of the surface 𝑑𝑆, and a fraction 𝑠 of the reflected light is reflected specularly, while a fraction (1−𝑠) is scattered from the collector surface in a nonspecular way [1]. After hitting the collector, specularly reflected light has direction 𝜎₁ as follows: 𝜎₁=𝜎−2(𝜎⋅𝑛)𝑛, (6) where 𝑛 is the internal normal unit vector to the surface at point 𝜉 given by 𝑥⃗𝑛=−𝑖+𝑦𝑗+2𝑓_𝑐𝑘𝑥²+𝑦²+4𝑓²_𝑐. (7)

In (7), 𝑖⃗, 𝑗, and 𝑘 are the 𝑂𝑥𝑦𝑧 coordinate orts. This fraction of light that is reflected specularly generates a force 𝑑𝑃_2𝑟=𝜌𝑟𝑠̃𝜎₁(𝜎,𝑛)𝑑𝑆, (8) while the nonspecularly reflected light generates a force given by 𝑑𝑃_2𝑠=𝜌𝐵_𝑓(1−𝑠)𝑟𝑛̃(𝜎,𝑛)𝑑𝑆. (9) Moreover, a portion of the absorbed light is reemitted backwards creating the force 𝑑𝑃_2𝑒(𝑟)̃𝜀=𝜌1−^𝑓𝐵_𝑓−𝜀_𝑏𝐵_𝑏𝜀_𝑓+𝜀_𝑏𝑛(𝜎,𝑛)𝑑𝑆. (10) In (10), 𝐵_𝑓 and 𝐵_𝑏 are the surface front and back non-Lambertian coefficients and 𝜀_𝑓 and 𝜀_𝑏 are the front and back emissivities. The total force applied is described as 𝑑𝑃₂=𝑑𝑃_2𝑟+𝑑𝑃_2𝑠+𝑑𝑃_2𝑒. (11) It produces the torque with respect to the center of mass 𝑂 given by 𝑑𝑀₂=𝜉×𝑑𝑃₂. (12)

Then, the light is reflected by the director whose orientation can be described by the normal to its surface 𝜈. One can assume that the light hitting D is only the fraction specularly reflected on the collector: as the director is small compared to the collector, the respective spatial angle observed from the collector is small and the influence of the scattered light may be disregarded. The incoming light from point 𝐴_𝑐 hits the director at point 𝐴_𝑑 with the position vector that can be found from condition (13) as follows: 0=𝜉_𝐴𝑑⋅𝜈=𝜉+𝜏𝜎₁⋅𝜈, (13) so, 𝜈𝜏=−𝜉⋅𝜎₁⋅𝜈,𝜉_𝐴𝑑=𝜈𝜉−𝜉⋅𝜎₁⋅𝜈𝜎₁. (14)

Force and torque transmitted by the light hitting the director are, respectively, as follows: 𝑑𝑃₃=−𝜌𝑟𝑠̃𝜎₁(𝑛)𝑑𝑀𝜎⋅𝑑𝑆,₃=𝜉_𝐴𝑑×𝑑𝑃₃=𝜌𝑟𝑠̃𝜎₁×𝜉(𝜎⋅𝑛)𝑑𝑆. (15)

The reflection of the light at the director produces force and torque that can be written, respectively, as 𝑑𝑃₄=𝜌𝑟𝑠̃𝜎₂(𝑛)𝑑𝑀𝜎⋅𝑑𝑆,₄=𝜉_𝐴𝑑×𝑑𝑃₄, (16) where 𝜎₂=𝜎₁−2𝜎₁⋅𝜈𝜈 (17) is the direction of the light reflected on the control mirror D.

The interaction of the light beam caught by the element 𝑑𝑆 of the collector C with the sail structure generates the following elementary force: 𝑑𝑃=𝑑𝑃₁+𝑑𝑃₂+𝑑𝑃₃+𝑑𝑃₄. (18)

The total force actuating on the sailcraft due to sunlight radiation pressure can be obtained by integration as follows: 𝑃=_𝐶𝑑𝑃. (19)

After integration, one gets 𝑃_𝑥=𝜋𝜌𝜒_𝑥+𝜁_𝑥𝑅ln1+^2𝑐4𝑓²_𝑐,𝑃_𝑦=𝜋𝜌𝜒_𝑦+𝜁_𝑦𝑅ln1+^2𝑐4𝑓²_𝑐,𝑃_𝑧=𝜋𝜌𝜒_𝑧+𝜁_𝑧𝑅ln1+^2𝑐4𝑓²_𝑐. (20)

Here, 𝜒_𝑥=16𝑓_𝑐𝜎_𝑥𝜆Ψ−6𝑓_𝑐𝑅²_𝑐×−𝜎_𝑥𝜎_𝑧+𝑟𝑠̃−𝜎_𝑥𝜎_𝑧1−2𝜈²_𝑥+2𝜈_𝑥𝜈_𝑧+𝜈_𝑦𝜎_𝑦𝜎_𝑧−2𝜈_𝑧𝜎²_𝑧,𝜒_𝑦=16𝑓_𝑐𝜎_𝑦𝜆Ψ−6𝑓_𝑐𝑅²_𝑐×−𝜎_𝑦𝜎_𝑧+𝑟𝑠̃−𝜎_𝑦𝜎_𝑧1−2𝜈²_𝑦+2𝜈_𝑦𝜈_𝑧+𝜈_𝑥𝜎_𝑥𝜎_𝑧−2𝜈_𝑧𝜎²_𝑧,𝜒_𝑧=4𝑓_𝑐𝜎_𝑧−2𝑓_𝑐+𝑅²_𝑐+4𝑓²_𝑐Ψ+𝑅²_𝑐𝜎²_𝑧+𝑟𝑠̃1−𝜎²_𝑧1−2𝜈²_𝑧−2𝜈_𝑧𝜈_𝑧+𝜈_𝑥𝜎_𝑥𝜎_𝑧+𝜈_𝑦𝜎_𝑦𝜎_𝑧, (21) with 𝜆=16𝑓³_𝑐+𝑅²_𝑐−8𝑓²_𝑐𝑅²_𝑐+4𝑓²_𝑐,𝐵Ψ=^𝑏𝜀_𝑏(𝑟̃−1)+𝐵_𝑓𝜀_𝑓+𝑟𝜀̃_𝑏−𝑟𝑠̃𝜀_𝑓+𝜀_𝑏𝜀_𝑓+𝜀_𝑏,𝜁_𝑥=8𝑓²_𝑐𝑟𝑠̃−𝜎_𝑥𝜎_𝑧1−2𝜈²_𝑥+𝜈_𝑥𝜈_𝑧+2𝜈_𝑦𝜎_𝑦𝜎_𝑧−3𝜈_𝑧𝜎²_𝑧,𝜁_𝑦=8𝑓²_𝑐𝑟𝑠̃−𝜎_𝑦𝜎_𝑧1−2𝜈²_𝑦+𝜈_𝑦𝜈_𝑧+2𝜈_𝑥𝜎_𝑥𝜎_𝑧−3𝜈_𝑧𝜎²_𝑧,𝜁_𝑧=4𝑓²_𝑐𝑟𝑠̃1−3𝜎²_𝑧1−2𝜈²_𝑧−2𝜈_𝑧𝜈_𝑧+2𝜈_𝑥𝜎_𝑥𝜎_𝑧+2𝜈_𝑦𝜎_𝑦𝜎_𝑧. (22)

For the case when the ideally reflecting SSPT is completely aligned with the sun direction (𝜎_𝑥=𝜎_𝑦=0 and 𝜎_𝑧=1) and the director rotates in the plane of the orbit (𝜈_𝑥=sin𝛼, 𝜈_𝑦=0, and 𝜈_𝑧=cos𝛼), expressions (20) are reduced to 𝑃_𝑋=Φ𝑟²_𝐸𝑟²_𝑠𝜋𝑅²_𝑐𝑓1−8^2𝑐𝑅²_𝑐𝑅ln1+^2𝑐4𝑓²_𝑐𝑃sin2𝛼,_𝑌𝑃=0,_𝑍=2Φ𝑟²_𝐸𝑟²_𝑠𝜋𝑅²_𝑐cos²𝑓𝛼−4^2𝑐𝑅²_𝑐𝑅ln1+^2𝑐4𝑓²_𝑐cos2𝛼. (23)

These expressions for the solar radiation force acting on SSPT have been used previously to study orbital transfer problems for SSPT and FSS [16, 22]. These studies show several advantages of SSPT scheme compared to FSS that are more evident for large-scale missions (i.e., for Solar System escape).

Calculation of the total torque due to the light pressure results in the following expressions: 𝑀_𝑠𝑥𝜎=𝜋𝜌^𝑦𝜐Ψ120𝑓²_𝑐+𝑅²_𝑐𝜎_𝑧4𝑓_𝑐𝑅²_𝑐−4𝑓²_𝑐𝜎_𝑦−4𝑓²_𝑐𝑟𝑠̃2𝜈_𝑥𝜈_𝑦𝜎_𝑥+1−2𝜈²_𝑥𝜎_𝑦,𝑀_𝑠𝑦−𝜎=𝜋𝜌^𝑥𝜐Ψ120𝑓²_𝑐+𝑅²_𝑐𝜎_𝑧4𝑓_𝑐−𝑅²_𝑐−4𝑓²_𝑐𝜎_𝑥+4𝑓²_𝑐𝑟𝑠̃2𝜈_𝑥𝜈_𝑦𝜎_𝑦+1−2𝜈²_𝑦𝜎_𝑥,𝑀_𝑠𝑧=2𝜋𝜌𝑓_𝑐𝑅²_𝑐𝑟𝑠̃𝜈_𝑧𝜎_𝑧𝜈_𝑥𝜎_𝑦−𝜈_𝑦𝜎_𝑥, (24) where Ψ is defined in (22) and 𝜐=64𝑓⁵_𝑐+3𝑅⁴_𝑐+4𝑓²_𝑐𝑅²_𝑐−32𝑓⁴_𝑐𝑅²_𝑐+4𝑓²_𝑐. (25)

These models for force and torque are used later on to study the attitude dynamics of SSPT.

3. SSPT Attitude Dynamics

Consider the problem of a SSPT attitude dynamics on a heliocentric orbit. This analysis is done using two right Cartesian reference frames located at the SSPT center of mass 𝑂. 𝑂𝑥𝑦𝑧 is the reference frame connected with the sailcraft; its axes are SSPT central principle axes of inertia. 𝑂𝑋𝑌𝑍 is the frame related to the orbital motion of the SSPT; its unit vectors along the axes are 𝑒_𝑍=𝑟_𝑂𝑟_𝑂,𝑒_𝑌=𝑒_𝑍×𝑣_𝑂𝑣_𝑂,𝑒_𝑋=𝑒_𝑌×𝑒_𝑍, (26) where 𝑟_𝑂 is the radius vector of point 𝑂 with respect to the center of the sun and 𝑣_𝑂 is SSPT’s center of mass velocity. Obviously, the sunlight follows in the direction of 𝑂𝑍 axis.

Mutual orientation of these frames can be described by the orthogonal matrix 𝐴=[𝑎_𝑖𝑗], where the elements 𝑎_𝑖𝑗 are the direction cosines of the axes of the 𝑂𝑥𝑦𝑧 system with respect to 𝑂𝑋𝑌𝑍. The transition from system 𝑂𝑋₁𝑋₂𝑋₃ to system 𝑂𝑥₁𝑥₂𝑥₃ can be done by three Euler’s rotations about axes 1, 2, and 3 through the angles 𝛾 (roll), 𝛼 (pitch), and 𝛽 (yaw). The direction cosines 𝑎_𝑖𝑗 are expressed through these angles as 𝑎₁₁=cos𝑥₁,𝑋₁𝑎=cos𝛼cos𝛽,₁₂=cos𝑥₂,𝑋₁𝑎=−cos𝛼sin𝛽,₁₃=cos𝑥₃,𝑋₁𝑎=sin𝛼,₂₁=cos𝑥₁,𝑋₂𝑎=sin𝛽cos𝛾+sin𝛼cos𝛽sin𝛾,₂₂=cos𝑥₂,𝑋₂𝑎=cos𝛽cos𝛾−sin𝛼sin𝛽sin𝛾,₂₃=cos𝑥₃,𝑋₂𝑎=−cos𝛼sin𝛾,₃₁=cos𝑥₁,𝑋₃𝑎=sin𝛽sin𝛾−sin𝛼cos𝛽cos𝛾,₃₂=cos𝑥₂,𝑋₃𝑎=cos𝛽sin𝛾+sin𝛼sin𝛽cos𝛾,₃₃=cos𝑥₃,𝑋₃=cos𝛼cos𝛾. (27)

At the heliocentric trajectories, the characteristic values of the angular acceleration due to orbital motion and gravity-gradient torque are much smaller than those created by cm-cp offset. Taking into account only the latter effect, the equations of attitude motion can be written as 𝐴𝑝̇+(𝐶−𝐵)𝑞𝑟=𝑀_𝑠𝑥,𝐵𝑞̇+(𝐴−𝐶)𝑟𝑝=𝑀_𝑠𝑦,𝐶𝑟̇+(𝐵−𝐴)𝑝𝑞=𝑀_𝑠𝑧, (28) where 𝐴, 𝐵, and 𝐶 are the principal moments of inertia of the sailcraft and 𝑀_𝑠𝑥, 𝑀_𝑠𝑦, and 𝑀_𝑠𝑧 are the components of the solar radiation torque given by (24). The projections of the sailcraft absolute angular velocity onto the axes 𝑂𝑥𝑦𝑧, 𝑝, 𝑞, and 𝑟 can be expressed as 𝑝=𝛾̇𝑎₁₁+̇̇𝛼sin𝛽,𝑞=𝛾𝑎₁₂+̇̇𝛼cos𝛽,𝑟=𝛾𝑎₁₃+𝛽̇. (29)

The direction of sunlight is 𝜎=(𝑎₃₁,𝑎₃₂,𝑎₃₃). Equations of motion (28) and (29) admit the following trivial solution: 𝛼=𝛽=𝛾=0. (30)

Solution (30) corresponds to the nominal equilibrium orientation. According to the SSPT force and torque model, the solar radiation torque acting on an SPT in the vicinity of this equilibrium has the components 𝑀_𝑠𝑥=𝐺𝜈_𝑥𝜈_𝑦𝛼+𝐹+𝜈²_𝑥𝐺𝑀𝛾,_𝑠𝑦=𝐹+𝜈²_𝑦𝐺𝛼+𝐺𝜈_𝑥𝜈_𝑦𝑀𝛾,_𝑠𝑧=𝐺𝜈_𝑧𝜈_𝑦𝛼+𝜈_𝑥𝛾, (31) where 𝐹=𝜋𝜌𝜐Ψ120𝑓²_𝑐+𝑅²_𝑐4𝑓_𝑐𝑅²_𝑐−4𝑓²_𝑐−𝑅²_𝑐𝑓_𝑐𝑟𝑠̃,𝐺=2𝜋𝜌𝑅²_𝑐𝑓_𝑐𝑟𝑠̃. (32)

Assuming that the attitude motion of SSPT occurs in the vicinity of the equilibrium orientation, (28) and (29) can be linearized, resulting in 𝐵𝛼̈−𝐹+𝜈²_𝑦𝐺𝛼−𝐺𝜈_𝑥𝜈_𝑦𝐴̈𝛾=0,𝛾−𝐺𝜈_𝑥𝜈_𝑦𝛼−𝐹+𝜈²_𝑥𝐺𝐶̈𝛾=0,𝛽=𝐺𝜈_𝑧𝜈_𝑦𝛼+𝜈_𝑥𝛾. (33)

Analyzing linearized system (33), one can notice that the motion of the 𝑂𝑧 axis does not depend on the yaw angle 𝛽. The rotation of the sailcraft about the collector symmetry axis is unstable. Considering that the sailcraft is an axisymmetric body, that is, 𝐴=𝐵, the stability conditions for motion of the 𝑂𝑧 axis can be written using the characteristic equation 𝐴²𝜆⁴−𝐴2𝐹+𝐺𝜈²_𝑥+𝜈²_𝑦𝜆²+𝐹𝐹+𝐺𝜈²_𝑥+𝜈²_𝑦=0. (34)

For stability of zero equilibrium position of system (33), characteristic equation (34) should have purely imaginary roots, which happens if and only if 2𝐹+𝐺𝜈²_𝑥+𝜈²_𝑦𝐹𝜈<0,𝐹+𝐺²_𝑥+𝜈²_𝑦𝜈>0,2𝐹+𝐺²_𝑥+𝜈²_𝑦²−4𝐹𝐹+𝐺𝜈²_𝑥+𝜈²_𝑦>0. (35)

As one can see, 𝐺>0, and so conditions (35) are equivalent to 𝐹<−𝐺𝜈²_𝑥+𝜈²_𝑦. (36)

𝑂𝑧 axis should be stable for all possible orientations of the director. Considering that director rotation is limited by condition (1), the components of the normal to the director surface 𝜈_𝑥 and 𝜈_𝑦 must be limited by 16𝑓²_𝑐𝑅²_𝑐4𝑓²_𝑐+𝑅²_𝑐²<𝜈²_𝑥+𝜈²_𝑦≤1. (37)

This condition is satisfied for all possible values of 𝜈_𝑥 and 𝜈_𝑦 if and only if 𝐹<−𝐺. (38)

Writing condition (38) in terms of the physical characteristics of the collector film, one gets ̃+115𝜓𝑟𝑠+𝜓−1𝜀_𝑏+𝜀_𝑓𝐵_𝑏𝜀_𝑏(𝑟̃−1)+𝐵_𝑓𝜀_𝑓+𝑟𝜀̃_𝑏−𝑠𝑟_𝑓𝜀_𝑏−𝑠𝑟_𝑓𝜀_𝑓×2+3𝜓−21+𝜓^3/2<0, (39) where 𝜓=(𝑅_𝑐/2𝑓_𝑐)².

3.1. Stability of a Solar Sail with Ideal Collector

For a sailcraft with an ideal collector, the surface optical parameters are [1] as follows: 𝑟̃=1,𝑠=1,𝜀_𝑏=𝜀_𝑓,𝐵_𝑏=𝐵_𝑓=23. (40) In this case, condition (39) becomes 4𝑓³_𝑐𝜋𝜌𝜓²<0. (41)

So, for all geometries of the sail, the 𝑂𝑧 axis is unstable.

3.2. Stability of a Solar Sail with Black Collector

For a sailcraft with an ideally black film in collector, the surface optical parameters are [1] 𝑟̃=0,𝑠=0,𝐵_𝑏=𝐵_𝑓=0. (42) In this case, condition (39) becomes 4𝑓³_𝑐𝜋𝜌𝜓𝜓−1<0. (43)

Condition (43) means that for an ideal black collector the motion of the 𝑂𝑧 axis is stable for 𝜓<1. So, for black sails with 𝑅_𝑐<2𝑓_𝑐, SSPT axis is stable.

3.3. Stability of a General Solar Sail

For sailcraft with general properties of the film, condition (39) holds for some 0<𝜓<𝜓₀. Considering the case described in [1], the film properties are 𝑟̃=0.88,𝑠=0.94,𝜀_𝑏𝜀=0.55,_𝑓=0.05,𝐵_𝑏=0.55,𝐵_𝑓=0.79. (44)

Condition (39) for this case means that the sailcraft 𝑂𝑧 axis is stable if 𝑅_𝑐<0.83𝑓_𝑐.

The stability of 𝑂𝑧 axis does not mean that the sailcraft is stable. Its motion about 𝑂𝑧 axis is always unstable, being a combination of harmonic oscillation and rotation with constant angular velocity. Meanwhile, the orientation of a SSPT about 𝑂𝑧 axis should be stabilized since it determines the director position, so a stabilizer for 𝛽 is always necessary.

4. Conclusion

In this paper, a new model for force and torque actuating on a compound solar sail is developed and used to study sailcraft motions. The attitude dynamics of a SSPT is studied for the case when the collector film possesses nonideal optical properties and the sailcraft symmetry axis is not aligned to the sun-sailcraft direction. Analysing the sailcraft stability, one can see that the solar radiation pressure destabilizes an ideal sail, but non-ideal collector film contributes to stabilization, resulting in stability, of its symmetry axis for all orientations of director when collector focal distance is large enough. However, an active control system is always necessary to guarantee stabilization of the sailcraft orientation with respect to the collector’s symmetry axis.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

This research is supported by the Brazilian program “Science without Borders” and Project ODySSea (PTDC/CTESPA/098030/2008) funded by the Portuguese Foundation for Science and Technologies (FCT), the Portuguese Operational Program for Competitiveness Factors (COMPETE), the Portuguese Strategic Reference Framework (QREN), and the European Regional Development Fund (FEDER).