Asteroid body-fixed hovering using nonideal solar sails

Transcription

1 Reseach in Aston. Astophys. 4 Vol. X No. XX, Reseach in Astonomy and Astophysics Asteoid body-fixed hoveing using nonideal sola sails Xiang-yuan Zeng, Fang-hua Jiang, Jun-feng Li School of Aeospace Engineeing, Tsinghua Univesity, Beijing 84, China; Coesponding jiangfh@tsinghua.edu.cn Received 4 May ; accepted 4 July Abstact: Asteoid body-fixed hoveing poblem using nonideal sola sail models in a compact fom with contollable sail aea is investigated in this pape. The nonlinea dynamic equations fo the hoveing poblem ae constucted fo a spheically symmetic asteoid. The feasible egion fo the body-fixed hoveing is solved fom the above equations by using a shooting method. The effect of the sail models, including the ideal, optical, paametic and sola photon thust, on the feasible egion is studied though numeical simulations. The influence of the asteoid spinning ate and the sail aea-to-mass atio on the feasible egion is discussed in a paametic way. The equied sail oientations and thei coesponding vaiable lightness numbes ae given fo diffeent hoveing adii to identify the feasibility of the body-fixed hoveing. An attactive mission scenaio is intoduced to enhance the advantage of the sola sail hoveing mission. Keywods: space vehicles: celestial mechanics---cosmology: obsevations INTRODUCTION CLOSE poximity missions to hazadous asteoids (Scheees 4) have been fequently investigated to be as a pecuso to some mitigation stategy o a contolled landing. Accoding to pevious studies, thee ae a numbe of possible options fo exploations in the vicinity of the asteoid, including Sun synchonous obit (Moow et al. ), etogade obit (Boschat & Scheees 5) and heliostationay flight (Moow et al. ) in which a spacecaft is placed in the system libation point (Baoyin & McInnes 5). Besides the above methods, the spacecaft could also maintain a equied fixed position elative to the otating asteoid efeed to be as the body-fixed hoveing. It is an effective way fo an asteoid human landing o a sample etun mission which has been successfully implemented by Hayabusa (Scheees 4). If the mission equies polonged obsevation of a specific aea locating away fom the synchonous obit, the thust must accommodate both the gavitational and centifugal foces. Thus, the extended hoveing peiod will highly depend on the onboad supplies of fuel fo chemical o continuous low-thust spacecaft. Compaed to the conventional spacecaft, the inheent capabilities of sola sailing without fuel consumptions make them well suited fo asteoid exploations. Close asteoid obital dynamics is quite challenging and complex due to thei iegula shape and otation (Hu & Scheees 8; Li et al. 3). Additionally, the sola adiation pessue (SRP) becomes a significant petubing foce in the vincinity of those small sized asteoids (Scheees 999). Anothe advantage of sola sailing is to utilize the SRP foce as an active contol. The fist detailed analysis about sail opeations at asteoids was made by Moow and Scheees (). Sawai () and Boschat and Scheees (5) investigated the body-fixed hoveing with conventional populsion systems. Zhang (3) extended such hoveing fom satellite-to-asteoid to the case of two satellites. Howeve, sola sail asteoid body-fixed hoveing was not addessed befoe Williams wok (9). In his wok (Williams & Abate 9), an ideally eflecting sail model associated with a sail efficiency facto (to eflect the diffeence between a tue sail and an ideal one) was adopted whose coesponding SRP foce is nomal to the sail suface. Nonideal sails have not

2 X.-Y. Zeng, F.-H. Jiang, J.-F. Li yet been discussed egading asteoid body-fixed hoveing although some heliostationay flight has eve been pesented by Moow () and Joba (). Sola sail has been seiously consideed as an altenative populsion system since the comet Halley endezvous mission. A numbe of demonstative missions (McInnes 999; Baoyin & McInnes 6) have been investigated along with thei coesponding pactical expeiments. Some damatic mission concepts involving non-kepleian obits (Gong et al. 7 & 9a; Vulpetti 997) have been poposed by using sola sailing. The successful flying of IKAROS and NanoSail-D has gained a lot of inteest fom the space community and laid a fist stone fo futhe sail missions (Gong et al. ). A concept of fulable sola sail was poposed by Williams (9) to genealize the body-fixed hoveing egion. Compaed to a fixed-aea sola sail with two vaiable attitude angles, the essence of a fulable sail is to sepaate the maximum SRP foce magnitude as an independent contol vaiable. Such a pefomance can be also implemented with a vaiable eflectivity sail film which has been patially demonstated on IKAROS to contol the sail attitude. In this pape a compact fom of nonideal sails (Mengali & Quata 7) with contollable sail aea is adopted to accomplish the asteoid body-fixed hoveing. A compaison is made to quantify the influence of the fou diffeent (including optical-, paametic-, and sola photon thust) sail models on the hoveing mission. Fo the taget asteoid, a spheically symmetic model is applied to be as an estimation of the fist step. The asteoid model can be elaxed and extended in futue studies. The analysis pesented hee complement the studies made by Williams (9) and extend to the scenaios with ealistic sails. Moeove, the effect of the asteoid otation and the sail aea-to-mass atio on the body-fixed hoveing egion is also illustated via numeical simulations. Finally, the sail contol pofiles coesponding to diffeent hoveing adii ae pesented to identify the feasibility of the body-fixed hoveing by using sola sailing. BODY-FIXED HOVERING FORMULATION. Equations of Motion In this analysis, a two-body gavitational model is adopted to descibe the dynamics of the spacecaft nea an asteoid. The vecto dynamical equation fo a sola sail in the unifomly otating body-fixed coodinate fame oxyz (Scheees et al. 998) can be witten as d d U a () SRP dt dt whee is the position vecto fom the asteoid cente of mass to the sailcaft, ω is the otational angula velocity vecto of the asteoid with espect to the inetial efeence fame IXYZ, U() is the gavitational potential of the asteoid and a SRP is the non-consevative sola adiation pessue (SRP) acceleation. The coodinate system IXYZ centeed on the asteoid is given in Fig.. The +IZ axis is along the diection of the asteoid angula velocity, the +IX axis is along the anti-sola diection and in the asteoid equatoial plane, and +IY axis is also in the asteoid equatoial plane making up an othogonal ight-handed tiad. Stictly speaking, the inetial fame IXYZ is a nea non-otating coodinate system due to the conic motion of the asteoid. Howeve, compaed to the asteoid otation peiod on the ode of hous to days at most, the otation of the fame IXYZ is on the ode of thousand days beyond AU away fom the Sun. Thus, fo those asteoids in the main belt, the fame IXYZ can be teated as an inetial efeence fame.

3 Asteoid Body-Fixed Hoveing using Nonideal Sola Sails e z Z Y e y n y Y Equatoial plane x sunlight I s e x Equatoial plane X Z z I o t X Fig. Obital efeence fames and sail attitude angles. The body-fixed fame oxyz coincides with the fame IXYZ at the initial time and the tansfomation matix fom oxyz to IXYZ is cos sin C t sin cos () whee the angle θ (t) = ωt. In ode to expess the SRP foce, an incident light coodinate system se x e y e z shown in Fig. is established whee the +se x axis is along the sunlight diection. The axis +se y coincides with the +IY axis and se z completes the ight-handed fame. In this fame, the unit vecto s diected fom the sun to the asteoid is always [,, ] T, which is the same as the unit vecto of the axis +se x. If thee is a sola latitude angle φ between the sunlight and the asteoid equatoial plane, the tansfomation matix fom IXYZ to se x e y e z is cos sin C,, (3) sin cos Seen fom Fig., if axis +se x is along axis +IZ coesponding to φ = π/, the Sun locates at the south pole of the asteoid. If φ = the sunlight is in the asteoid equatoial plane.. Sola Sail Foce Model A unified, compact fom of sola sails with fixed sail aea has been pesented by Mengali and Quata (7) to accomplish the advanced heliostationay missions. Compaed to the ideal sail with a pefectly flat eflective suface, the optical model takes the effect of eflection, absoption and eadiation into account. The paametic model consides the billowing of the sail. Additionally, the sola photon thust (SPT, detailed by Gueman et al. 9) is also included in the model. The sail acceleation with vaiable sail aea can be witten as a s t sun p q s 3q q t cos qb x qbbcos b3cos s R e n (4) AU whee the supescipt s indicates that the vecto is expessed in the se x e y e z fame. In the above equation, μ sun is the sola gavitational constant (.3744 m 3 s - ) and R AU is the Sun-asteoid heliocentic distance in the unit of AU ( AU.496 m). The coefficients [p, q, b, b, b 3 ] coesponding to diffeent sail models ae specified in Table (but see discussions by Mengali & Quata 7). Table Sola sail foce model coefficients p q b b b 3 3

4 X.-Y. Zeng, F.-H. Jiang, J.-F. Li Ideal Optical Paametic SPT The sail cone angle α shown in Fig. is defined as the angle between the sail nomal vecto n and the incident light e x ( e s x = [,, ] T ) fo both ideal and optical models. Howeve, fo the paametic and SPT sails, α is the angle between the SRP foce and the vecto e x. The sail oientation can be explicitly expessed as cos, s n sinsin, (5) [, ) sincos whee δ is the clock angle shown in Fig., defined as the angle between the pojected line of the incident light onto the plane se y e z and axis +se z. Fo an ideal sail, Eq. (4) becomes sun t t cos s s a n (6) RAU whee the sail lightness numbe β(t) is the atio of the SRP acceleation to the local sola gavitational acceleation, which only depends on the sail aea-to-mass atio σ(t) SL SL t, max (7) m A t t whee the citical sail loading paamete σ SL is a constant whose value is appoximately.53 g m -, m is the total mass of the sailcaft and A(t) is the effective eflective suface of the sail. Hee, β max is the maximum available lightness numbe which is key design paamete fo a mission..3 Body-Fixed Hoveing The body-fixed hoveing leads to a fixed equilibium point in the fame oxyz at a desied position. Fom Eq. (), the enabling SRP acceleation should be Ux x U asrp U y y U z (8) z The desied position can be expessed by the latitude angle λ [-π/, π/] and the longitude angle θ [, π] as x cos cos y cos sin (9) z sin whee is the magnitude of the position vecto. It is assumed hee that the asteoid is spheically symmetic. Thus, the gavitational acceleation exeted on the sailcaft is U x cos cos U ast ast U y cos sin 3 () U z sin whee μ ast is the gavitational constant of the asteoid. Substituting Eq. (9) and Eq. () into Eq. (8), the sail acceleation is given as T ast ast ast a SRP coscos, cossin, sin () 4

5 Asteoid Body-Fixed Hoveing using Nonideal Sola Sails A constaint of the sola sail is that the SRP foce can be only poduced in the anti-sola hemisphee. Theefoe, the sail acceleation must satisfy T s T SRP s s SRP a ex C C t a e x () Substituting Eq. () and Eq. (3) into Eq. (), one can obtain s T s ast ast asrp e x coscoscos sin sin (3) whee θ(t) = θ + θ (t) = θ + ωt. Since the body-fixed hoveing of a sailcaft coves the whole peiod of the asteoid otation, the angle θ(t) takes all values fom to π. In ode to guaantee the value of Eq. (3) to be always positive, the ight second tem ast sin sin should be always positive in that cosθ [-, ]. It indicates that the angles λ and φ must be of the same sign. Accoding to the definitions of these two angels, the coesponding situation is that the sailcaft and the Sun must lie in diffeent sides of the asteoid equatoial plane. To accomplish the body-fixed hoveing, the equied SRP acceleation in Eq. () should be the same as that povided in Eq. (4): s s s asrp a t C C tasrp a t (4) Thee ae thee contol vaiables (β, α, δ) coesponding to the above thee dimensional nonlinea equations. It seems impossible to obtain analytical solutions but could be solved numeically. Fo a specified position as expessed in Eq. (9), if thee ae a set of time-vaiant (β, α, δ) making Eq. (4) zeo when θ takes all values of to π, the hoveing point is feasible and vise vesa. The nonlinea equations can be solved using the Matlab s fsolve function with a default method of dogleg. In ode to impove the calculation efficiency, a pogam of MinPack- tanslated into C++ language is adopted to solving the nonlinea equations. In all simulations, the toleance of Eq. (4) is satisfied to be bette than 9. Since Eq. (4) is only thee dimensional without integations, it is not sensitive to the initial values. Duing the simulations pe unning when θ takes all values of to π, thee ae some bad cases needed to be veified coesponding to feasible solutions (i.e. < β β max, α π/). Specifically, the contol attitude angles must be in thei feasible domains. Thee have been two cases aisen in ou simulation pocess which can be tansfomed into feasible solutions. The two cases and thei equivalent expessions ae (,] [, ), (5) mod, [,] mod, whee the function mod is the modulus afte division. 3 CASE STUDY The effect of sail coefficients, the asteoid otating angula velocity and the sail aea-to-mass atio on the feasible egion of the body-fixed hoveing will be examined in this section. The influence of the hoveing adius on the sail contol histoy is investigated though numeical simulations. The main paametes of the asteoid consideed hee ae the same as given by Williams (9). The heliocentic obit of the asteoid is assumed to be cicula at.7 AU. Its diamete is. km with a density of.4 3 kg m -3 and its otational peiod is 9. hous with spin axis aligned with axis +oz. The highest sola lightness numbe is.53 coesponding to the aea-to-mass atio of g m -. The sola latitude angle is set to be 6 deg fo such a main-belt asteoid. Except special explanations, all simulations in this section ae in the above paametes. 3. Effect of Sola Sail Foce Coefficients Figue shows the feasible body-fixed hoveing egions with fou diffeent sail models whose coefficients ae aleady given in Table. Since the asteoid is assumed to be spheically symmetic, feasible egions fo θ fom to π should be the same. Those feasible egions locate at the asteoid s nothen hemisphee since the Sun is below its equatoial plane (φ = 6 deg). The egion of the SPT sail is lagest while the smallest is paametic. Feasible egions of the optical and paametic sails ae nealy the same although thei model coefficients ae totally diffeent. 5

6 X.-Y. Zeng, F.-H. Jiang, J.-F. Li Ideal Optical Paametic SPT.5 oz / (km) ox / (km) Fig. Feasible egions of fou diffeent sail models. Fo those fou sail models, all egions stat fom the coesponding synchonous points in the equatoial 3 plane at a adius of syn (, hee syn.3 km). When the sailcaft eaches the asteoid s syn ast nothen pole, all SRP foce is used to countebalance the asteoid gavitational foce. Fo an ideal sail, it is easy to identify that the closest hoveing adius is ast min (6) R max sun AU sin whose value in ou case is appoximately.95 km. The value of min fo optical and paametic sails is.3 km while that is.88 km fo the SPT sail. As a ough estimation, an efficiency facto of.85 can be added to Eq. (6) to appoximate the optical and paametic models hee (see esults of Williams 9). The shadow effect of the asteoid fom the Sun is neglected which educes the effective hoveing egion fo such a spheical model. Fo othe iegula shaped asteoids, such an effect needs to be discussed in detail. 3. Asteoid Rotational Effects Figue 3 illustates the effect of asteoid spinning ate on the hoveing egions with the asteoid at the lowe cente. Two spin angula velocities ae consideed, i.e., 9 hous and 5 hous (which is abitaily selected longe than 9 hous). Only the ideal sail model is adopted hee and that s why the minimum anti-sun pole adius min is the same at.95 km. Fo the hypothetical case of slow otating the synchonous obit adius is appoximately.84 km. Thee is ovelapping aea between the two egions when the hoveing latitude angle λ is geate than.5π. The feasible hoveing egion of the lowe ω is a little lage than the highe case. It indicates that thee ae moe options of hoveing positions (but with elatively fathe hoveing adius fo the same latitude) about slow otating asteoids compaed to those of simila physical popeties and obital paametes. 6

7 Asteoid Body-Fixed Hoveing using Nonideal Sola Sails 5h 5h.5 9h 9h oz / (km) Fig ox / (km) Effect of asteoid otation on the feasible hoveing egion. 3.3 Effect of the Aea-to-Mass Ratio In this section, the effect of the sail aea-to-mass atio σ(t) on the hoveing adius will be investigated. Accoding to ecent studies, a chaacteistic acceleation on the ode of.5 mm s - can accomplish nea-tem sail missions while a elatively mid-tem 6-m squae sail has been envisaged by NASA fo the Sola Pola Image (SPI) mission (Mengali & Quata 9). The chaacteistic acceleation a c is defined as the maximum sail acceleation at AU when the sail nomal diection is along the sunlight diection. Thus, the lowe bounday value of σ min is g m - coesponding to a nea-tem sail with a c =.9 mm s -. The uppe bounday of σ min is 4 g m - whose a c is appoximately.7 mm s -. Fo a sail with vaiable sail aea, the minimum aea-to-mass atio σ min coesponds to the highest sail lightness numbe based on Eq. (7). The investigated values of σ min (β max, a cmax ) have been given in Table along with thei coesponding minimum hoveing adii. These fou values ae enough to illustate the influence of σ min on the feasible hoveing egions. Additionally, the minimum hoveing adius fo σ min = 4 g m - is.599 km which is only m away fom the asteoid suface. Thee is no need to be close fo an obsevation mission fo such a fictitious asteoid. Table Minimum hoveing adius fo diffeent values of the aea-to-mass atio σ min / (g m - ) β max a cmax / (mm s - ) min / (km) Figue 4 shows the vaiation of the hoveing adius with espect to the hoveing latitude in tems of each aea-to-mass atio. The cuves above the synchonous obit on.3 km ae the oute boundaies of each feasible egion while below cuves ae inne boundaies. The body-fixed hoveing adius is obtained with the vaiation of the hoveing latitude in a step of.π. It is easy to know that the feasible egion of σ min = 4 g m - is lage than the othe thee cases due to its highe chaacteistic acceleation. The biggest gap of hoveing adius between diffeent values of σ min in Fig. 4 occus at the asteoid pola egion in a value of.35 km fom Table. 7

8 X.-Y. Zeng, F.-H. Jiang, J.-F. Li 4 Hoveing adius / (km) Feasible Syn Hoveing latitude / (ad) Fig. 4 Relation between hoveing adius and latitude fo diffeent sail aea-to-mass atios The body-fixed hoveing fo diffeent latitudes with espect to the asteoid is feasible by using sola sailing. When the hoveing latitude is specified, the incease of σ min in the above scenaios can incease a little of the hoveing boundaies, especially fo the aeas away fom the pola egion. Fo example, if the hoveing latitude λ is.64π, the inne bounday fo σ min = 4 g m - is.94 km while. km fo σ min = g m -. The loss of 7 m hoveing height fo σ min = g m - makes its system mass.5 times than the case of σ min = 4 g m -. It indicates that thee is a tadeoff between the hoveing adius and the system payload mass. Taking the 6-m squae sail of the demonstated SPI mission as an example, the payload mass fo σ min = g m - is 56 kg. On the contay, the payload mass fo σ min = 4 g m - is only kg. Theefoe, it is pefeable to take moe payload mass fo low-pefomance sails to accomplish the body-fixed hoveing mission. 3.4 Effect of the Hoveing Radius The effect of the hoveing adius on the sail contol pofile will be examined in this section. Without loss of geneality, the hoveing latitude is set to be π/4 and the minimum sail aea-to-mass atio is g m -. The bounday values fo this hoveing latitude ae 6 m and 5 m, espectively. Thee hoveing adii ae investigated, i.e., 56 m, 36 m and 456 m whee 36 m is the synchonous adius and the two othes in the feasible egion ae 5 m away fom the synchonous obit. Since the incident light fame IXYZ is assumed to be non-otating, the sail contol pofile fo the body-fixed hoveing is symmetical with espect to the IXZ plane. Let s assume thee ae n + discete points of the contol pofile between to π whee n is an intege. In the cuent simulations, n is 5 coesponding to the calculating step of.π fo the asteoid otation. Then the contol vaiables afte π can be obtained as n n nn (7) n n The sail contol pofile in the fist half peiod is shown in Fig. 5 fo each case, including the sail lightness numbe and the two sail attitude angles. To distinguish the sail oientations in the bottom plotting of Fig. 5, the clock angle fo the outwad case (456 m) is given as a segmented figue. Fo all these thee cases, the peak of the lightness numbe does not exceed the maximum value of.53. Fo both the inwad (56 m) and outwad cases, the lightness numbe and the sail attitude angles ae time-vaiant to fulfill the equiement of the non-synchonous body-fixed hoveing. 8

9 Asteoid Body-Fixed Hoveing using Nonideal Sola Sails Lightness numbe Asteoid otational angle / (ad) in syn out Attitude angles / (ad) Fig in 7 6 out 5 3 syn Asteoid otational angle / (ad) Sola sail contol pofile in the fist half peiod fo diffeent hoveing adii. in syn out It is inteesting that the clock angle is always zeo fo the synchonous case (36 m with λ = π/4) while the contol vaiables β and α ae constant. It indicates that the equied sail acceleation is only povided to countebalance the asteoid gavitational acceleation along the axis IZ. Such a condition can be explicitly deduced fo an ideal sail fom section II. Fo a body-fixed hoveing position at a synchonous height out of the equatoial plane, the equied sail acceleation in Eq. () is simplified into T ast a syn,, sin (8) syn Substituting Eq. (5) into Eq. (6) with δ =, one can obtain cos s sun a t t cos R (9) AU sin In ode to guaantee the feasibility of the body-fixed hoveing, the povided sail acceleation of Eq. (9) should be equal to that equied of Eq. (8). Theefoe, the sail cone angle must satisfy α = π/ - φ to make sue that the sail acceleation is along the +IZ diection. In such a case, the equied lightness numbe can be obtained as ast RAU sin syn sun syn sin () Fo the above case, its sail lightness numbe can be calculated fom Eq. () as.57 which is consistent with the value shown in Fig. 5. It is a good popety that the asteoid body-fixed hoveing can be accomplished by using sola sailing with constant contol vaiables at the synchonous adius out of the equatoial plane. Such a popety would be vey attactive fo futue sola sailing body-fixed hoveing missions. Fist, the self-otating peiod of the asteoids is diffeent fom each othe due to the lage numbe of asteoids. Second, the sail attitude contol fo a lage thin film is vey challenging and complicated (Wie & Muphy 7; Gong et al. 9b). Theefoe, fo specified hoveing latitudes out of the asteoid equatoial plane, the sailcaft placed on the height of the synchonous obit can maintain the fixed position with constant attitudes and lightness numbe. Fo the hypothetical spheical asteoids, the petubed gavitational acceleations fom othe celestial 9

10 X.-Y. Zeng, F.-H. Jiang, J.-F. Li bodies ae also ignoed in this study. In fact, they may play a key ole in the stability of the hoveing obit. Moeove, the vaiation of the Sun-asteoid distance due to the asteoids eccentic obits should have a geat influence on the dynamics of the hoveing obit (Faes & Joba ). All above effects should be taken into account in futue studies. The stability of these hoveing obits and thei contollability will be the subject of next publishing wok. In tems of divese shapes of the asteoids, a hoveing study aound elongated asteoids is in pogess based on the cuent famewok. 4 CONCLUSIONS Feasible egions of asteoid body-fixed hoveing have been investigated by using sola sailing. Fou diffeent sail models with the vaiable sail aea ae consideed including the ideal, optical, paametic and SPT sails. Nonlinea equations ae constucted to obtain the feasible hoveing egion and solved with a shooting method. With advanced thust ability, the SPT sail can poduce the lagest hoveing egion which shinks to smalle ones fo othe sails. Fo the bounday hoveing adius of both inwad and outwad cases, the effect of the nonideal models must be consideed in the mission design. The asteoid spinning ate plays a key ole on the hoveing egion. With same physical and obital chaacteistics, an asteoid with a highe value of otating peiod (hee is 5 hous coesponding to a slow otating asteoid) holds a lage hoveing egion than the one with shote peiod (9 hous). Fo a desied feasible hoveing position away fom the synchonous obit, both the sail lightness numbe and its attitude have to be adjusted to countebalance the asteoid otation and the levitated gavitational acceleation out of the equatoial plane. Fo the hoveing positions on the height of the synchonous obit out of the equatoial plane, the sail acceleation is only povided to countebalance the levitated gavitational acceleation esulting in the constant sail attitude and lightness numbe. Such a popety is vey attactive fo sola sail body-fixed hoveing missions and will be extended to diffeent shaped asteoids in futhe studies. Acknowledgments This wok was suppoted by the National Basic Reseach Pogam of China (973 Pogam, CB7) and Tsinghua Univesity Initiative Scientific Reseach Pogam (No ). Thank D. Jin-guang Li fo his help about the Latex. Refeences Boschat, S. B., Scheees, D. J., 5, JGCD, 8 (), 343 Baoyin, HX., McInnes, C. R., 5, JGCD, 8 (6), 38 Baoyin, HX., McInnes, C. R., 6, JGCD, 9 (3), 538 Faés, A., Joba, À.,, Poc. 63th IAC, Italy (IAC-.C.6.4) Gong, SP., Baoyin, HX., Li, JF., 7, JGCD, 3 (4), 48 Gong, SP., Li, JF., Baoyin, HX., 9a, CMDA (Celestial Mechanics and Dynamical Astonomy), 5 (-3), 59 Gong, SP., Baoyin, HX., Li, JF., 9b, AA (Acta Astonautica), 65 (5-6), 73 Gong, SP., Li, JF., Gao, YF.,, RAA, (), 5 Gueman, A. D., Sminov, G. V., and Peeia, M. C., 9, Mathematical Poblems in Engineeing, 9, Hu, WD., Scheees, D. J., 8, ChJAA, 8(), 8 Li, XY., Qiao, D., Cui, PY., 3, ASS (Astophysics and Space Science), 348, 47 McInnes, C. R., 999, Sola Sailing: Technology, Dynamics and Mission Applications, st edition, (Spinge) Moow, E., Scheees, D. J., and Lubin, D.,, JSR, 38(), 79 Moow, E., Scheees, D. J., and Lubin, D.,, AIAA, 4994, Mengali, G., Quata, A. A., 7, AA, 6, 676 Mengali, G., Quata, A. A., 9, JSR, 46, 34 Scheees, D. J., Osto, S. J., and et al., 998, ICARUS, 3(), 53 Scheees, D. J., 999, JAS, 47, 5 Scheees, D. J., 4, AIAA, 445, Sawai, S., Scheees, D. J., and Boschat, S.,, JGCD, 5(4), 786 Vulpetti, G., 997, AA, 4, 733 Wie, B., Muphy, D., 7, JSR, 44(4), 89 Williams, T., Abate, M., 9, JSR, 46(5), 967 Zhang, JR., Zhao, SG., and Yang, YZ., 3, IEEE AES, 49(4), 74