The Journal of the Astronautical Sciences, Vol. 54, No. 1, January–March 2006, pp. 17–27

Orbital Dynamics of SunFacing Solar Sails Under the Constraint of Constant Sail Temperature1 Hiroshi Yamakawa2

Abstract The orbital dynamics of Sun-facing solar sails is investigated considering a constraint of constant sail temperature at the limit of the sail material. Although solar sails can normally be articulated so as to provide thrust with both a transverse and radial component, a Sunfacing attitude with the center of solar pressure behind the center of gravity may be preferred for very large or gossamer sails in order to achieve Sun-facing attitude stability. The proposed Sun-facing solar sails are applicable to space weather and geo-storm warning missions for monitoring the inner solar system environment by in-situ measurement of solar wind plasma and high-energy particle events. Constraining the temperature of the sail to the temperature limit of the sail material allows the innermost circular orbits to be attained thereby maximizing scientific returns. The stability of the heliocentric circular orbit under such radial thrust with the constant temperature constraint is investigated, and the stability conditions are obtained as functions of the radius of circular orbit and the solar sail lightness number accounting for optical/thermal properties.

Introduction The concept of utilizing large sheets of reflective material to harness radiation pressure from the Sun has remained a theme in literature since the days of the early space pioneers. Solar sailing has been considered in a diverse range of future space missions, and recent advances in lightweight materials have now made this technology realistic. The present paper focuses on the orbital dynamics of Sun-facing solar sails under the constraint of constant sail temperature at the maximum temperature limit. The maximum temperature limit of the solar sail material must be taken into account for any mission that involves travel through interplanetary space 1

Presented as paper AAS 2005-362 at the AAS/AIAA Astrodynamics Specialists Conference, Lake Tahoe, CA, August 7–11, 2005. 2 Department of Space Systems and Astronautics, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Kanagawa 228-8510 Japan. Email: [email protected]. Also, Research Institute for Sustainable Humanosphere, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan. Email: [email protected]. 17

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within Earth’s orbit. Koblik et al. [1] conducted a numerical investigation of locally optimal spacecraft trajectories for a planar solar sail based on travel from Earth orbit to near-Sun space, where the maximum temperature represents a severe constraint on sail applications. Dachwald [2] similarly considered the temperature limit of the sail film in the investigation of optimal trajectories for solar sail missions to the outer planets and into near interstellar space involving one or more close approaches to the Sun. In the present study, the orbital behavior of Sun-facing solar sails in near-Sun space is investigated assuming a constraint that the sail temperature be maintained constant. To maximize scientific returns, the constant sail temperature is set at the maximum temperature limit of the sail material, thereby allowing trajectories involving the innermost circular orbits. The proposed Sun-facing solar sails are thus applicable to space observatories in the inner solar system environment. As an example of such a deployment, West [3] investigated a geo-storm warning mission stationed Sunward of the L1 point of the Sun-Earth system. The National Institute of Information and Communications Technology (NICT) and the Japan Aerospace Exploration Agency (JAXA) [4, 5] proposed an L5 mission for space weather research and operational forecasting experiment, involving the deployment of a spacecraft at the L5 point of the Sun-Earth system. Such a mission would provide remote sensing of the Sun and the interplanetary space, as well as in-situ measurements of the solar wind plasma and high-energy solar particle events. The L1 and L5 points are thus candidate locations for space weather missions. However, observations closer to the Sun and at variable distances may also dramatically enhance scientific returns. The Sun-facing solar sail system proposed in the present study would allow in-situ measurement to be performed at various heliocentric distances by retaining the operability of the sail. If the target orbital radius is fixed, the sail may be dropped once the target orbit has been achieved. However the sail has to be operative even after the initial orbit has been reached, since the space weather mission requires the sail orbit control capability to change the heliocentric distance. The present treatment considers only the behavior at heliocentric circular orbits, and does not deal with the transfer from the Earth to the initial heliocentric circular orbit or transfers among circular orbits. Solar sails typically have the ability to be articulated so as to provide thrust with both a transverse and radial component. However, a Sun-facing attitude with the center of solar pressure behind the center of gravity may be preferred for very large or gossamer sails, providing autonomous Sun-facing attitude stability. Some form of attitude control thruster system is assumed for initial attitude acquisition. The total acceleration is thus constrained to the radial direction. The classical problem of spacecraft trajectory under continuous radial acceleration has been investigated by many researchers. For example, Broucke and Akella [6] described the general types of solutions for the continuous constant outward radial acceleration problem, employing numerical integration and concepts such as the theory of periodic orbits and Poincare’s characteristic exponents as a basis. Broucke [7] reviewed several results related to the classical problem of two-dimensional particle motion in a field with a central force proportional to a real power of the distance. The present discussion deals with the stability of the heliocentric circular orbit under radial thrust assuming Sun-facing solar-sails given a constant sail temperature constraint. A stability index, defined as the sum of the two roots of the characteristic equation of the monodromy matrix, is also investigated as a means of evaluating new periodic orbits reachable by bifurcation from the stable circular orbits.

Orbital Dynamics of Sun-Facing Solar Sails Under Constant Sail Temperature

19

Temperature Model A non-ideal solar sail possesses non-perfect reflectivity, that is, the reflection coefficient rref is less than unity. The radiation input to the sail is thus governed by the thermal property 1  rref, and also by the pitch angle  of the solar sail relative to the Sun line (i.e., angle between the Sun-sail line and the tilted sail normal). The sail temperature can be obtained from the balance between the thermal input 共1  rref兲W cos , where W is the solar flux, and the heat emitted from the sail 共f  b兲SBT 4, where f and b are the front and back emissivities and SB is the Stefan-Boltzmann constant 共5.671 10 8 Wm2K 4兲. The sail temperature may thus be written as T苷

冋

共1  rref兲W cos  SB共f  b兲

冉冊

W 苷 WE

RE r

册

1/4

2

where WE 共苷1,368 W兾m2兲 is the energy flux at 1 AU (i.e., at the mean radius of Earth orbit R E 苷 1.496 10 11 m) [8], and r is the heliocentric distance of the solar sail. The variables hereafter are normalized to the radius of Earth’s orbit, the corresponding circular velocity, and the magnitude of the gravitational acceleration of the Sun at 1 AU. The period of Earth’s orbit is 2, and the reference time unit is 58.1 days. The reference distance, velocity, and acceleration units are 1.496 10 11 m, 29.8 km兾s, and 0.00593 m兾s2, respectively. For a constant temperature, the relation between the pitch angle  and the heliocentric distance r is then finally given by cos  苷 r 2

(1)

where

苷

f  b SBT 4 1  rref WE RE2

(2)

Equation (1) poses a constraint on the heliocentric distance, as given by

r 2  1

(3)

In the present treatment, the thermal/optical properties take values of rref 苷 0.88, f 苷 0.05, and b 苷 0.55 ([8], p. 50; [9], p. 228). Using these values,  苷 5.31 共AU兲2 is obtained from equation (2) assuming T 苷 400 K. This result is employed as a baseline. It should be noted that this discussion is valid only in near-Sun space (i.e., r  ⵑ0.43 AU ) as constrained by equation (3).

Equations of Motion The sail is usually assumed to be a flat, mirror-like specular reflector with negligible loss, and the total radiation pressure force in this case is normal to the back surface of the sail. For the purpose of the present treatment, the Sun is regarded as a point-like source of radiation. The orbital dynamics are handled assuming the same ideal force model with rref 苷 1. Realistic factors, such as the differences in force magnitude and direction (see reference [8], Figs. 2.9 and 2.10), are not considered in this ideal case. As a spacecraft system, a combination of large flat tiltable solar sails is considered for the provision of radial thrust (Fig. 1). The sail system consists of

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Yamakawa

FIG. 1. Force Model for Sun-Facing Solar Sails.

two reflective sheets mounted on a retractable truss, which maintains the sail system structure and controls the pitch angle by adjustment of the truss length. This Sun-facing solar sail suffers a loss of efficiency due to a cosine-squared reduction in the magnitude of the solar radiation pressure force and cosine-cubed reduction in the radial component as the pitch angle increases. The heliocentric equation of motion in the radial direction for such a flat tilted solar sail may be written using equation (1) in plane polar coordinates 共r, 兲 as

冉冊

d2r d r 2 dt dt

2

苷

1 1 1   2 cos3 苷  2   3r 4 2 r r r

(4)

Here,  is the polar angle of the solar sail measured counterclockwise from some reference position ([8], p. 118), and  is the dimensionless sail lightness number, which is defined as the ratio of the solar radiation force to the solar gravitational force acting on the solar sail, given by

 苷 *兾

Orbital Dynamics of Sun-Facing Solar Sails Under Constant Sail Temperature

21

where  is the sail loading parameter (i.e., solar sail mass per unit area) and * is the critical solar sail loading parameter (i.e., sail loading such that sail acceleration is equal to the acceleration of solar gravity). Note that the right-hand side of equation (4) becomes a function of radial distance alone. The critical loading parameter * is given by * 苷 L s兾共2 GMs c兲 关kg  m2兴 Given parameters Ms (solar mass, 1.989 10 30 kg), G (universal gravitational constant, 6.672 10 11 m3kg1s2), c (speed of light, 2.998 10 8 m兾s) and L s (solar luminosity, L s 苷 4R E2WE) ([8], p. 40), this equation gives * 苷 1.53 10 3 kg  m2. The characteristic acceleration a c, which is an equivalent design parameter to the solar sail loading, may be written assuming an ideal force model as a c 苷 2P兾 关m  s2兴 where P is the solar radiation pressure magnitude at 1 AU, which has a value of 4.56 10 6 N  m2.

Orbit Stability It is assumed for the subsequent discussion that the sail system consists of two reflective sheets tilted with respect to the direct Sun direction, as applicable for large or gossamer sails. The total acceleration of the solar sails is constrained to the radial direction, allowing the stability of the circular orbit solution to be evaluated in terms of linearized equations. The conservation of angular momentum, h, and energy, C, can be expressed in terms of the potential energy V共r兲 [10] r2 1 2

d 苷h dt

冋冉 冊 冉 冊 册 dr dt

2

 r2

d dt

2

 V共r兲 苷 C

Changing the variable from the heliocentric distance, r, to the inverse distance, u, using the relation u 苷 1兾r gives [10] 1 2

冋冉 冊 册 du d

2

 u2 

V共r兲 苷 C h2

Further, assuming an attractive force f of f 苷 dV共r兲兾dr the equation of motion [10] in the radial direction can be derived as d2u W共u兲 2  u 苷 d h2

(5)

where W共u兲 苷 f兾u2 This result can be used to investigate circular orbits. The heliocentric distance of the circular orbit is derived by setting d2u兾d 2 in equation (5) to zero, that is u0 苷

W共u 0兲 f 苷 2 2 2 h u0h

(6)

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Yamakawa

The angular momentum can then be determined given an initial inverse radius. Linearization around the circular orbit is performed by setting u 苷 u0  x where 兩x兾u 0兩  1. Substituting this into equation (5) gives d2x W共u 0  x兲  u0  x  苷0 2 d h2 and

冉

冊

1 dW d2x  u 0  x  2 W共u 0兲  x 苷0 2 d h du u苷u0 after Taylor expansion retaining only the first-order terms at small displacement x. This result leads to d 2x  m2x 苷 0 d 2

(7)

where m2 苷 1 

u 0 dW W共u 0兲 du u苷u0

Thus, the solution consists of a simple harmonic oscillation (m2  0) and an exponential function (m2  0). Introducing the inverse radial distance u 苷 1兾r, equation (4) can be rewritten as 1   3u6 d 2u  u 苷 d 2 h2 where a positive acceleration on the right-hand side corresponds to attractive acceleration. The functions f and W, and the derivative of W, are obtained as f 苷 u2   3u4 W共u兲 苷 1   3u6 dW 苷 6 3u7 du Therefore, the coefficient m2 becomes m2 苷

u 60  7 3 u 60   3

the denominator of which is positive from equation (6). This relation states that for Sun-facing solar sails with radial acceleration undergoing a cosine-cubed reduction with increasing pitch angle, the stability around circular orbits can be divided into the two regions

 3  u 60  7 3 (m2  0, unstable) u 60  7 3 (m2  0 stable)

(8) (9)

Orbital Dynamics of Sun-Facing Solar Sails Under Constant Sail Temperature

23

These two equations state that the frequency and stability around a circular orbit are both functions of the optical/thermal () and physical () properties of the solar sail, as well as the radius of the reference circular orbit. Figure 2 plots the stability region of circular orbits with respect to heliocentric distance and solar sail lightness number, where  苷 5.31 共AU兲2 and T 苷 400 K. Note that Fig. 2 is specific to this particular temperature constant. The heliocentric radius should be smaller than 0.434 AU, otherwise the temperature would not reach 400 K from equation (3). The pitch angle  can be calculated by equation (1), given the orbital radius and the optical/thermal property . The pitch angle is zero at the maximum orbital radius (i.e., 0.434 AU in Fig. 2), and increases with proximity to the Sun. The boundary between the stable and unstable regions can be derived from equations (8) and (9). As the sail moves closer to the Sun, the allowable maximum lightness number for stable motion increases, resulting in a corresponding decrease in the minimum solar sail mass per unit area. If the sail is pointed directly at the Sun, simple orbit mechanics dictate that the sail orbit will not be stable. This can be understood from a conventional case with  苷 0 in the central photogravitational field of the Sun (i.e., solar gravity and radiation).

FIG. 2.

Stability Regions for Sun-Facing Solar Sails 共 苷 5.31 共AU兲2, T 苷 400 K兲.

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Yamakawa

In this case, equation (4) can then be written as d 2u 1 2  u 苷 d h2 and the functions f and W and the derivative of W become f 苷 共1  兲u2 W共u兲 苷 1   dW兾du 苷 0 Thus, m 苷 0, indicating that the orbit around a circular orbit in the central photogravitational field of the Sun is not stable.

Stability Index The stable case is considered here in more detail. The state transition matrix H共兲 of equation (7) is given by normalizing such that H共0兲 is the identity matrix, as given by H共0兲 苷

冋

cos m m sin m

共1兾m兲 sin m cos m

册

The columns of H共兲 are the two linearly independent solutions of equation (7). The monodromy matrix, that is, the matrix H共2兲 evaluated at the end of a complete revolution ( 苷 2 on the nominal circular orbit), is given by H共2兲 苷

冋

cos m2 m sin m2

共1兾m兲 sin m2 cos m2

册

The eigenvalues may be obtained as the roots of the characteristic equation

冋

Det

cos m2   m sin m2

册

共1兾m兲 sin m2 苷0 cos m2  

which reduces to the simple form

2  2 cos m2  1 苷 0 These expressions indicate that the product of the two roots is unity, and that the roots are in fact reciprocals of one another. The two eigenvalues 共1, 2兲 may be real or complex. Real eigenvalues denote an unstable orbit, while complex conjugates on the unit circle yield a stable orbit. The sum of the two roots is referred to as the stability index (k), and represents a convenient basis for the discussion of stability since the index is real in both the stable and unstable cases. This is expressed by the relation k 苷 1  2 苷 2 cos m2 The stable case corresponds to k 僆 关2, 2兴. Broucke and Akella [6] investigated the constant radial thrust problem using a stability index with the same definition, and showed that the frequency of the linearized orbit (m) can become the inverse of an integer depending on the radius of the corresponding circular orbit. This special circular orbit lies at the origin of a new family of periodic orbits with period equal to an integer times the length of the circular orbit. Following the method employed in

Orbital Dynamics of Sun-Facing Solar Sails Under Constant Sail Temperature

25

reference [6], bifurcation to a new periodic orbit with n times as many revolutions occurs when the angle is equal to 2兾n 共n  1兲 Here, four simple cases are considered: n 苷 2, m 苷 1兾2, k 苷 2.0 (double period) n 苷 3, m 苷 1兾3, k 苷 1.0 (triple period) n 苷 4, m 苷 1兾4, k 苷 0.0 (quadruple period) n 苷 6, m 苷 1兾6, k 苷 1.0 (six-fold period) Figure 3 plots the stability index as a function of the heliocentric distance of the circular orbits for Sun-facing solar sails, assuming a lightness number of  苷 0.2 (i.e., a c 苷 1.19 10 3 m兾s2, and  苷 7.65 10 3 kg  m2 at T 苷 400 K). For the selected parameters, Sun-facing solar sails allow double, triple and quadruple period bifurcations. A double period orbit, for example, begins to repeat its track after two revolutions in two periods of the original circular orbit. Figure 4 shows the results of a numerical simulation involving a bifurcation orbit (a double-period orbit). The nonlinear equation of motion (4) is used for numerical integration. This example shows that the stability index provides a good picture of bifurcation phenomena around stable circular orbits. These bifurcation orbits can also be applied

FIG. 3.

Stability Index and Period Bifurcation for Sun-Facing Solar Sails 共 苷 5.31 共AU兲2, T 苷 400 K,  苷 0.2兲.

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Yamakawa

FIG. 4. Double-Period Orbit for a Sun-Facing Solar Sail 共 苷 5.31 共AU兲2,  苷 0.2, 共r兲 苷0 苷 0.391, 共dr兾dt兲 苷0 苷 0.03兲.

for space weather missions enhancing the flexibility of the orbit selection for in-situ measurement.

Conclusions A Sun-facing solar-sail system that is applicable for space weather missions and that allows relocation of a deployment at various heliocentric distances was proposed in which the orbital dynamics is constrained with respect to maintaining constant sail temperature at the sail temperature limit of the sail material. The linear stability of the heliocentric circular orbit under radial thrust conditions was investigated, and stability conditions were obtained in terms of the radius of circular orbit (r  ⵑ0.43 AU) and the lightness number of the solar sail, given optical/thermal properties ( 苷 5.31 共AU兲2) and a constant temperature of 400 K. A stability index, defined as the sum of the two roots of the characteristic equation of the monodromy matrix, was investigated as a means of evaluating new periodic orbits reachable by bifurcation from the stable circular orbits. It was shown that the stability index provides a good picture of bifurcation phenomena around stable circular orbits.

Orbital Dynamics of Sun-Facing Solar Sails Under Constant Sail Temperature

27

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