Displaced Geostationary Orbit Design. Using Hybrid Sail Propulsion

Transcription

1 Diplaced Geotationary Orbit Deign Uing Hybrid Sail Propulion Jeannette Heiliger *, Matteo Ceriotti, Colin R. McInne and Jame D. Bigg Univerity of Strathclyde, Glagow G1 1XJ, Scotland, United Kingdom Abtract Due to an increae in number of geotationary pacecraft and limit impoed by eat-wet pacing requirement, the geotationary orbit i becoming congeted. To increae it capacity, thi paper propoe to create new geotationary lot by diplacing the geotationary orbit either out of or in the equatorial plane by mean of hybrid olar ail and olar electric propulion. To minimize propellant conumption, optimal teering law for the olar ail and olar electric propulion thrut vector are derived and the performance in term of miion lifetime i aeed. For comparion, imilar analye are performed for conventional propulion, including impulive and pure olar electric propulion. It i hown that hybrid ail outperform thee propulion technique and that out-of-plane diplacement outperform in-plane diplacement. The out-of-plane cae i therefore further invetigated in a pacecraft ma budget to determine the payload ma capacity. Finally, two tranfer that enable a further improvement of the performance of hybrid ail for the out-ofplane cae are optimized uing a direct peudo-pectral method: a eaonally tranit between orbit diplaced above and below the equatorial plane and a tranit to a parking orbit when geotationary coverage i not needed. Both tranfer are hown to require only a modet propellant budget, outweighing the improvement they can etablih. * Ph.D. Candidate, Advanced Space Concept Laboratory, Department of Mechanical Engineering Reearch Fellow, Advanced Space Concept Laboratory, Department of Mechanical Engineering, AIAA Member Director, Advanced Space Concept Laboratory, Department of Mechanical Engineering, AIAA Member Aociate Director, Advanced Space Concept Laboratory, Department of Mechanical Engineering Page 1 of 41

2 Nomenclature a A c1, c 2 Thrut induced acceleration vector Area Contant g Earth tandard gravitational acceleration at urface, m/ 2 h iobl Out-of-plane diplacement ditance Obliquity of the ecliptic I p Specific impule J k L m Cot function Specific performance Miion lifetime Ma m gimbal Gimbal ma m P SEP thruter power ource ma m pay Payload ma m prop Propellant ma m tank SEP propellant tank ma n ˆn N P r t T t p u U V Integer Solar ail acceleration unit vector Number of impule per orbit Power Poition vector Time SEP thrut vector Time in between impule Control vector Effective potential Velocity Page 2 of 41

3 W Solar energy flux denity at 1 AU, 1367 W/m 2 x State vector Pitch angle Solar ail lightne number Sun incidence angle Yaw angle d Ditance between pacecraft and GEO m Change in ma r t In-plane diplacement Time interval V Intantaneou velocity change Efficiency In-plane angle (cylindrical/pherical coordinate) Gravitational parameter of central body Projected orbital radiu (cylindrical coordinate) Sytem loading * Critical ail loading, 1.53 g/m 2 ω Orbital period Out-of-plane angle (pherical coordinate) Angle in Earth orbital plane meaured from winter oltice Angle between r and equatorial plane Angular velocity vector Orbital angular velocity of a circular Keplerian orbit with r rnko Subcript At time t t C E In Earth fixed inertial reference frame In Earth fixed rotating reference frame Page 3 of 41

4 GEO i f max min NKO r,, R S SA SEP TF Referring to geotationary orbit At final time At i th node Maximum Minimum Referring to non-keplerian orbit Referring to pherical coordinate component In rotating reference frame Referring to olar ail Referring to Sun Referring to olar array Referring to olar electric propulion Referring to thin film olar cell Supercript - Prior to impulive V + After impulive V * Optimal ˆ Firt order derivative Second order derivative Unit vector Introduction The geotationary orbit (GEO) hot atellite for telecommunication and Earth obervation. With an orbit period equal to the Earth rotational period, pacecraft in GEO are tationary with repect to an oberver on the Earth, allowing for a continuou downlink to terretrial communication uer. But, with only one uch unique orbit, the GEO ha become congeted over time, epecially above the continent where concentration of geotationary atellite are greatet [1]. Page 4 of 41

5 In order to create new geotationary lot, thi paper invetigate the ue of diplaced non-keplerian orbit (NKO) to diplace the GEO either above/below or in the equatorial plane. Such diplaced NKO can be generated by applying a continuou, thrut-induced acceleration to counterbalance or augment part of the local gravitational acceleration [2]. The exitence, tability and control of diplaced NKO have been tudied for both the two- and three body problem [3-5] and numerou application have been propoed. Thee application range from pacecraft proximity operation [6] to NKO diplaced high above the ecliptic to enable imaging and communication for high latitude [7] and diplaced NKO for lunar far ide communication and lunar outh pole coverage [8-9]. Solar ail have often been propoed a pacecraft propulion ytem to maintain diplaced NKO [2, 4, 7-8, 1]. Solar ail exploit the radiation preure generated by photon reflecting off a large, highly reflecting ail to produce a continuou, propellant-le thrut [2]. Although the concept of olar ailing ha been conidered for many year, only recently ha a mall ail been uccefully deployed by the Japanee demontrator miion IKAROS and by NASA NanoSail-D nanoatellite [11-12]. Depite thi advance, the Technology Readine Level (TRL) of olar ailing a primary propulion ytem on a reaonable ized miion i till rather low. That, in combination with a high advancement degree of difficulty [13] and the inability to generate a thrut component in the direction of the Sun [2] limit the application of olar ailing. Solar electric propulion (SEP) ha alo been conidered a a mean to maintain diplaced NKO [9, 14]. SEP ue the acceleration of ion to produce a relatively low thrut, but enable high pecific impule. It ha flown on multiple miion including Deep Space 1 (1998), SMART-1 (23), Dawn (27) and GOCE (29), reulting in a high TRL and a low advancement degree of difficulty. Neverthele, the application of SEP are alo limited due to a bound on the available propellant ma. Conidering the diadvantage and limitation of olar ail and SEP, ome author are uggeting to hybridize the two ytem, becaue the eparate ytem complement each other: ince only mall olar ail will be required, the hybridization lower the olar ail advancement degree of difficulty. Furthermore, while the olar ail lower the demand on the SEP propellant ma, the SEP ytem can provide a thrut component in the direction of the Sun (which the olar ail i unable to generate). Thi i under the aumption that the SEP ytem i mounted on a gimbal uch that the two propulion ytem can teer independently of each other. Hybrid ail have already been uggeted to enable interplanetary tranfer [15-16], to allow for periodic orbit in the vicinity of the Lagrange point in the Earth-Moon ytem for lunar communication purpoe [17], and to generate artificial equilibria in the Earth-Sun Page 5 of 41

6 three-body problem [18-2]. All tudie how to ome extent an improvement for hybrid ail over the ue of pure SEP or pure olar ailing in term of propellant ma conumption, required thrut magnitude level and/or initial pacecraft ma. In thi paper, we propoe the ue of hybrid ail to enable out-of-plane and in-plane diplaced GEO, thereby extending initial reearch conducted for out-of-plane diplaced GEO in Ref. [21]. Furthermore, compared to the olar ail levitated GEO propoed in Ref. [1] and [22], the diplaced GEO propoed in thi paper will allow pacecraft to be tationary with repect to their ground tation, becaue the reidual in-plane acceleration that caue a drift in the pure olar ail cae can be cancelled by the SEP thruter. Alo, diplacement well beyond the geotationary tation keeping box will be enabled uing relatively mall, near-term olar ail. After a brief introduction in Section I of the general theory underlying two-body diplaced NKO uing continuou control, the out-of-plane and in-plane diplaced GEO a invetigated in thi paper will be defined. Uing thee definition, Section II and III will derive the performance of impulive and pure SEP control for maintaining the diplaced GEO in term of propellant conumption and miion lifetime. Similar reult will be obtained for the hybrid ail cae in Section IV by deriving the SEP and olar ail optimal teering law. Since the out-of-plane cae outperform the in-plane cae, the out-of-plane cae will be ued for a detailed ma budget analyi in Section V to ae the performance in term of payload ma capacity. Finally, two type of tranfer that improve the performance of the out-of-plane diplaced GEO will be optimized for the SEP propellant conumption by olving the accompanying optimal control problem in Section VI and reult in term of SEP propellant ma will be preented. I. Diplaced GEO Diplaced GEO, or diplaced NKO in general, can be found by eeking equilibrium olution to the two-body problem in a rotating frame of reference. A tranformation to an inertial frame will ubequently how that the pacecraft execute a circular orbit diplaced away from the nominal Keplerian orbit [3]. Fig. 1 how thee reference frame where,, R x y z i a rotating frame of reference that rotate with contant angular velocity R R R ω = ˆ z R with repect to an inertial frame I( X, Y, Z ), where the z R axi and Z axi coincide. Furthermore, to maintain the diplaced NKO a thrut-induced acceleration a i aumed. The equation of motion of the pacecraft in the rotating reference frame are then given by: Page 6 of 41

7 r 2ωr U a (1) with the gravitational parameter of the central body and U the effective potential that combine the gravitational potential of the central body and a potential that repreent the centripetal acceleration: 1 2 U ( / r) ωr (2) 2 Equilibrium olution can ubequently be found by etting r r in Eq. (1), eliminating the firt two term: U a (3) which directly give the magnitude and direction of the thrut acceleration required to maintain the diplaced NKO. Diplaced NKO Z z R a ω r h Spacecraft y R Y X x R Fig. 1 Diplaced non-keplerian orbit reference frame. Becaue ω i contant, no tranvere component of the thrut can exit, requiring the thrut vector to lie in the plane panned by the radiu vector and the vertical axi. The thrut direction i therefore defined by the pitch angle only: tan zˆ R zˆ R U U (4) Finally, the potential U can be written uing a et of cylindrical polar coordinate (,, h) a hown in Fig. 1: U 2 1 ( ) 2 / r (5) Subtituting Eq. (5) into Eq. (4) and (3) reult in the following required thrut direction and magnitude to maintain the diplaced NKO: 2 tan (, h; ) 1 (6) h Page 7 of 41

8 2 a(, h; ) h (7) * * with the orbital angular velocity of a circular Keplerian orbit with a radiu equal to the radiu of the NKO: 3/2 * r h (8) A. Out-of-plane diplaced GEO While the expreion derived above hold for any two-body diplaced NKO, it i given for the diplaced GEO 3 that GEO / rgeo, with rgeo km. Furthermore, auming a diplacement h to create an outof-plane diplaced GEO, Eq. (7) can be ued to find the correponding projected radiu that minimize the required acceleration. Taking the firt derivative of Eq. (7) with repect to and etting it equal to zero yield the following condition: r r 3 h r (9) The complex and negative real root of thi ixth order polynomial are ignored and Decarte Rule of Sign i applied to find that Eq. (9) ha one ign change and therefore one poitive real root [23]. An analytical olution to Eq. (9) wa not found, therefore a numerical method in the form of Newton method wa applied [24]. The reult for a large range of out-of-plane diplacement are illutrated in Fig. 2, which how the acceleration contour plot for GEO and include the olution to Eq. (9). The figure how the correctne of the approach a the olution connect the extrema of the eparate contour line, i.e. the minimum acceleration required to provide a particular outof-plane diplacement. The figure furthermore how that, clearly, the maller the out-of-plane diplacement, the maller the required acceleration. However, for the diplaced GEO, the minimum diplacement i predefined by the geotationary tation keeping box to prevent the pacecraft from interfering with other atellite in the GEO. Combining International Telecommunication Union (ITU) regulation and regulation drawn up by individual countrie uch a by the US Federal Communication Commiion (FCC), a geotationary tation keeping box of can be defined, equaling a box ize of km centered around the geotationary atellite [25-26]. Thi lead to a range of diplacement ditance of km, where 36.8 km repreent the cae that the pacecraft jut hover above Page 8 of 41

9 the GEO tation keeping box, while the higher diplacement alo take a tation keeping box for the diplaced pacecraft into account. Three diplacement ditance will therefore be conidered in thi paper, namely 35, 75 and 15 km. Solving Eq. (9) for thee three diplacement and ubequently uing Eq. (6) to find the required thrut acceleration pitch angle, provide the optimal diplaced GEO a defined in Table 1. From the pitch angle it become clear that an almot pure out-of-plane acceleration (i.e. ) i required. Eq. (6) ubequently how that, to obtain, the condition * hould be atified, which in it turn require r rgeo. Subtituting thi into Eq. (6) and (7), give the required thrut direction and magnitude to maintain uch a diplaced GEO: tan (1) h a h (11) 2 * 3 rgeo Thi type of orbit correpond to a o-called Type I NKO, which i table for modet diplacement [2]. A chematic of thi type of out-of-plane diplaced GEO i provided in Fig. 3. Contrary to the cae in Table 1, the Type I diplaced GEO allow for an analytical derivation of the performance of hybrid ail control, and will therefore be ued in thi paper for the out-of-plane cae. Epecially ince the difference in acceleration with repect to the minimized acceleration given in Table 1 i only percent at maximum (i.e. for h 15 km) and will therefore only reult in a lightly conervative etimate of the performance. Fig. 2 Acceleration contour plot for GEO including the minimized acceleration for a given out-of-plane diplacement h. The acceleration i dimenionle with repect to the gravitational acceleration at unit planet radiu and i marked on the contour. Page 9 of 41

10 Table 1 Definition of minimum acceleration out-of-plane diplaced GEO. h, km, km, deg a, mm/ 2 ± ± ± Out-of-plane Z z R a y R X GEO In-plane r GEO r GEO r h a x R Y Fig. 3 Definition of out-of-plane and in-plane diplaced GEO. B. In-plane diplaced GEO Rather than diplacing the GEO out-of-plane, another option would be to diplace the GEO in-plane, i.e. in the equatorial plane, ee Fig. 3. Subtituting h into Eq. (6) and (7) provide the thrut direction and magnitude required to maintain uch an in-plane diplaced GEO: 1 (12) GEO * a (13) with 3 * and rgeo r, where the radial diplacement r and r for orbit diplaced outide or inide the GEO, repectively. Eq. (12) furthermore how that a pure radial thrut i required, directed either inward ( ) or outward ( ), depending on the ign of r, to increae or decreae the angular velocity of a Keplerian orbit with radiu to the angular velocity of the GEO. An initial aement of the relative performance of the out-of-plane and in-plane diplaced orbit can be derived from the contour plot in Fig. 2. The figure how that for mall and equal diplacement in out-of-plane (along the vertical axi) and in-plane (along the horizontal axi) direction, the ratio of the required acceleration i Page 1 of 41

11 approximately 3, a can alo be hown from Hill equation [24]. The in-plane diplacement thu require an acceleration three time higher than an equally diplaced out-of-plane orbit. The acceleration contour furthermore how that it i lightly more advantageou to diplace the orbit outide ( r ) the GEO than inide: for the ame acceleration a larger diplacement outide than inide the GEO can be achieved. Thi paper will therefore alway conider the r cae (a depicted in Fig. 3) for the in-plane diplaced GEO. II. Impulive control Although a continuou acceleration i required to achieve a diplaced NKO, impulive control uing a chemical propulion ytem can be employed to maintain a minimum diplacement from a Keplerian orbit. By providing multiple impulive velocity change along the diplaced GEO, the pacecraft can bounce on the diplaced orbit. Then, at time t the pacecraft i located at the diplaced GEO and an intantaneou change in velocity, or impule, V, i given. Thi will caue the pacecraft to lightly move away from the diplaced GEO. However, ince no thrut i applied in between pule, the pacecraft follow a natural Keplerian orbit after the impule, cauing the pacecraft to cro the diplaced GEO after ome time. Upon thi croing, another impule i given to revere the pacecraft velocity and tart the cycle again. Thi concept i illutrated in Fig. 4 for an out-of-plane diplaced GEO. The ue of impulive control to maintain diplaced NKO ha been invetigated before and ha among other been uggeted to hover above Saturn ring [3, 27] and to maintain a local cluter of pacecraft for high reolution imaging of terretrial or atronomical target uing interferometry technique [28]. V V V Z V V V Diplaced GEO ω GEO z Spacecraft r Y X GEO P x y Fig. 4 Illutration of impulive control for an out-of-plane diplaced GEO and definition of reference frame for Hill equation. Page 11 of 41

12 For mall diplacement, the required magnitude of the impule can be computed uing the linearized Hill equation that repreent the dynamic of a pacecraft in the vicinity of a point P on a circular Keplerian reference orbit, ee Fig. 4 [3, 24]. A detailed derivation i given in Ref. [3] and i therefore not repeated here. Only the reult are provided. For the firt impule the following hold: 2 3GEO xt p tan GEOt p 2 Vx, x (14) 3 t 8tan t 2 GEO p GEO p GEOtp 2tan GEOtp 2 Vy, y 6xGEO (15) 8tan t 2 3 t GEO p GEO p V tan 2 z, z GEO z GEOt p (16) with x and z defined in the rotating reference frame hown in Fig. 4 and tp N, where i the orbital period and N the number of impule per orbit. For the out-of-plane diplaced GEO x( t ) x (), y( t ) y() and z( t ) z() and therefore only repeated impule in x and z direction have to be provided, but with double the p p p required V to revere the direction of the velocity vector. For an in-plane diplaced GEO x( t ) x (), p y( t ) y() and z( t ) z (), which require only repeated, double magnitude impule in x direction. p p Fig. 5a how the reult for an out-of-plane diplaced GEO levitated 35 km above the equatorial plane for one orbital revolution and for different number of impule along the orbit. The top plot in Fig. 5b furthermore how the V required to maintain uch an orbit, while the bottom plot provide imilar information for a 35 km in-plane diplaced orbit. The figure clearly how the larger V required for the in-plane cae than for the out-of-plane cae, a expected from the analyi in Section I.B. It alo how that a higher number of pule i advantageou when diplacing the orbit out-of-plane, i.e. the penalty on the V due to puled rather than continuou control become le. Contrary, for the in-plane diplaced GEO, a higher number of impule i diadvantageou, becaue, although the amount of V per impule decreae, the decreae i not ufficient to compenate for the larger number of impule that need to be provided. The analyi in Fig. 5 can be extended from one orbital revolution to multiple revolution to obtain the performance of impulive control for maintaining the diplaced GEO in term of miion lifetime. Thi lifetime, L, i defined a the epoch at which a particular ma fraction, mf / m, i obtained: Page 12 of 41

13 m m ( m m ) m (17) f prop with m f, m and m prop the final, initial and propellant ma, repectively. The propellant ma can be computed through an iterative approach and uing the rocket equation that give the ratio of the ma prior ( m ) and after ( m ) the impule providing the combined V : V Ip g m e m (18) with I p the pecific impule and g the Earth gravity contant. Eq. (18) in combination with the reult in Fig. 5b immediately how that the out-of-plane diplaced GEO will outperform the in-plane diplaced GEO due to the maller amount of V required. The reult, a hown in Fig. 6, are therefore only provided for the out-of-plane cae. Fig. 6 conider both a range of pecific impule (from current to near term and far-future technology) and a range of ma fraction for the three diplacement ditance of 35, 75 and 15 km. Furthermore, 1 impule per orbit are aumed to provide a balance between the complexity of the miion, the penalty on the V for puled control and the deviation from the nominal diplaced orbit. Note that the ymmetry of the problem caue the reult for GEO diplaced above and below the equatorial plane to be exactly the ame. a) b) Diplaced GEO GEO Fig. 5 (a) 35 km out-of-plane diplaced GEO with impulive control (b) Required V for a 35 km out-ofplane (top) and in-plane (bottom) diplaced GEO for different number of impule per orbit N. Page 13 of 41

14 a) b) c) h = ±35 km h = ±75 km h = ±15 km Fig. 6 Out-of-plane diplaced GEO maintained with impulive control ( N 1 ): miion time L a a function of the pecific impule I p and the ma fraction mf / m, for different value of the diplacement ditance h. The graph in Fig. 6 can be interpreted in different way. For example, for a 35 km diplaced GEO, an average pecific impule of 32 [29] and a ma fraction of.5 a lifetime of.36 year can be achieved. Conidering a lifetime of 1-15 year for current geotationary pacecraft, Fig. 6 how that imilar lifetime cannot be achieved uing impulive control. Only a maximum of 1.9 year can be obtained for the mallet diplacement ditance and for extreme value of the pecific impule and the ma fraction. The caue of thi poor performance lie in the penalty on the V for puled rather than continuou control and the low pecific impule of chemical propulion ytem. III. SEP control Thi ection invetigate the ue of SEP to maintain the diplaced GEO in order to improve the performance of the diplaced GEO with repect to the ue of impulive control. The performance of SEP control in term of miion lifetime for a particular ma fraction can be aeed by conidering the following differential equation for the ma: T m (19) I g p with T the SEP thrut magnitude. Since the required acceleration i contant (ee Eq. (7)), the lifetime can be derived analytically from Eq. (19). Subtituting T a m into Eq. (19) and rearranging give: mf tf dm a dt m I g (2) m t p Page 14 of 41

15 Evaluating thee integral and etting t yield the following lifetime: L m I g tf ln m f p a (21) Eq. (21) how that, clearly, a higher required acceleration reduce the miion lifetime. Conidering the fact that the in-plane diplaced GEO require a larger acceleration than an equally diplaced orbit out-of-plane (ee Section I.B), a horter lifetime can be expected for the in-plane cae. The reult, a hown in Fig. 7, are therefore again only provided for the out-of-plane cae. Fig. 7 how the miion lifetime for an arbitrary initial ma, a wide range of pecific impule and ma fraction and for the three diplacement ditance of 35, 75 and 15 km. Again, the reult hold both for GEO diplaced above the equatorial plane and for thoe diplaced below the equatorial plane, due to the ymmetry of the problem. a) b) c) h = ±35 km h = ±75 km h = ±15 km Fig. 7 Out-of-plane diplaced GEO maintained with SEP control: miion time L (a maximum of 15 year i conidered) a a function of the pecific impule I p and the ma fraction mf / m, for different value of the diplacement ditance h. The graph can be interpreted imilarly to the graph in Fig. 6. Comparing Fig. 7 with Fig. 6 immediately how a dramatic improvement of the lifetime for an SEP controlled pacecraft over an impulive controlled pacecraft. Again, conidering a ma fraction of.5 and auming a currently feaible SEP pecific impule of 32 (e.g. a flown on the Hayabua pacecraft [3]), the lifetime i increaed from 4.3, 2. and 1. month for impulive control to 3.7, 1.7 and.9 year for 35, 75 and 15 km diplaced orbit, repectively. However, lifetime of 1-15 year a Page 15 of 41

16 for current geotationary pacecraft can till only be achieved for the mallet diplacement of 35 km and either for low ma fraction (e.g. mf.1 and Ip 26 ) or for high pecific impule (e.g. mf.45 and Ip 75 ). IV. Hybrid ail control To improve the performance of the diplaced GEO even further, thi ection will invetigate the ue of hybrid ail control. For thi, the acceleration required to maintain the diplaced GEO, a, ee Eq. (1) or equivalently Eq. (1) and (11) for the out-of-plane cae and Eq. (12) and (13) for the in-plane cae, i written a the um of the acceleration generated by the SEP ytem, a SEP, and the acceleration produced by the olar ail, a : a a a (22) SEP To maximize the lifetime of the miion, the objective i to minimize the magnitude of the acceleration required from the SEP ytem: min asep min where the acceleration generated by the olar ail i given by: S 2 r ˆ ˆ 2 a a (23) a n r n ˆ (24) with S the gravitational parameter of the Sun. Note that an ideal, i.e. a perfectly reflecting, ail i aumed. The unit vector in the direction of the olar radiation preure force, ˆn, i therefore directed normal to the ail urface. Furthermore, the magnitude of the Sun-ail vector, r, i approximated by a contant Sun-Earth ditance of 1 AU. The parameter i the olar ail lightne number and can be defined a the ratio of the olar radiation preure acceleration and the olar gravitational acceleration, or equivalently a the ratio of the ytem loading (i.e. the ratio of * the pacecraft ma to the olar ail area, m/ A ) and the critical ail loading, 1.53 g/m 2 [2]: * (25) Eq. (25) how that for a ail loading equal to the critical ail loading, the lightne number i unity, indicating that the olar radiation preure acceleration i exactly equal to the olar gravitational acceleration. The equation furthermore how that the ail lightne number i a function of the pacecraft ma. Since the ma of the hybrid Page 16 of 41

17 ail pacecraft decreae due to the conumption of propellant by the SEP ytem, the parameter increae according to: where the ubcript indicate the tart of the miion. m (26) m Due to the tilt of the Earth rotational axi with repect to the ecliptic plane, the direction of the Sun-ail unit vector r ˆ change during the year. To model thi variation, an Earth fixed rotating reference frame E( xe, ye, ze ) a hown in Fig. 8 i ued. Centered at the Earth with the ( xe, y E) plane in the equatorial plane and the z E axi along the rotational axi of the Earth, thi reference frame rotate with the ame angular velocity a the Earth in it orbit around the Sun, cauing the unit vector r ˆ to alway be contained in the ( xe, z E) plane. The angle decribe the poition of the Earth along it orbit (with = at winter oltice), while the angle i defined a the angle between r ˆ and the equatorial plane and i therefore a function of. Thi angle i at it maximum at winter oltice ( () iobl ) and at it minimum at ummer oltice ( ( ) iobl ) with i obl the obliquity of the ecliptic. Summer z E y E Spring rˆ z E z E x E rˆ rˆ z E x E rˆ y E Winter Autumn z E rˆ ( ) i obl z E SEP ˆn aˆ SEP y E x E i obl x E SEP Fig. 8 Definition of reference frame and parameter ued to model the eaonal variation of rˆ and to define the olar ail and SEP pitch and yaw angle. Page 17 of 41

18 The variation of i in magnitude equal to the olar declination, but i oppoite in ign: yielding: 1 ( ) in in i obl co (27) co ˆ r (28) in The unit vector normal to the ail urface, ˆn, can be decribed uing the ame frame of reference, ee Fig. 8. Uing the olar ail pitch angle,, and yaw angle,, the unit vector ˆn i given by: in co ˆ n inin (29) co Subtituting Eq. (29) and the expreion for r ˆ, ˆn and into Eq. (22) and rearranging give: a a a a m (co in co S SEP, xe xe 2 m r a 2 in co ) in co m (co in co S SEP, ye ye 2 m r a 2 in co ) in in m (co in co S SEP, ze ze 2 m r 2 in co ) co (3) with a x E, a y E and a z E the component of the acceleration required to maintain the diplaced GEO a defined in Eq. (1) and (11) for the out-of-plane cae and in Eq. (12) and (13) for the in-plane cae. For intance, for the out-ofplane cae, a x E a y E and a ze h r 3 GEO. In that cae, the SEP ytem thu need to counterbalance the in-plane component of the olar ail acceleration and need to augment the out-of-plane olar ail acceleration to obtain the required out-of-plane acceleration. Inpecting Eq. (3) how that for a given value for m and (i.e. for a particular intant of time), the minimization problem in Eq. (23) i merely a function of the olar ail pitch and yaw angle and therefore reduce to finding the optimal olar ail pitch and yaw angle that minimize the acceleration required from the SEP ytem: Page 18 of 41

19 , arg min a (, ) (31),min,,max, * * SEP where the domain of i defined later in the paper. The next two ubection olve thi minimization problem for the out-of-plane and in-plane cae eparately. A. Out-of-plane diplaced GEO For the out-of-plane cae, the olution to Eq. (31) can be found by etting the partial derivative of the SEP acceleration with repect to the ail pitch and yaw angle equal to zero: a SEP a SEP (32) Performing thi analyi for the yaw angle yield: a SEP nˆ rˆ nˆ rˆ 4c c c co co in in 2 (33) with m h c, c (34) S m r rgeo For Eq. (33) to hold throughout the year and conidering that c1 and ( nr ˆˆ ) (to generate a olar ail acceleration) the optimal yaw angle equal: n (35) * with n an integer equaling either or 1. Subtituting thi value into Eq. (3) (with a a and x E y E a ze 3 GEO h r ) how that the y E component of the SEP thrut force i zero at all time. Furthermore, conidering the fact that the olar ail i unable to generate a thrut component in the direction of the Sun and recalling that the x E axi point away from the Sun at all time, Eq. (35) can be reduced to: (36) * A imilar analyi can be performed for the partial derivative with repect to the ail pitch angle. Subtituting give the following condition: * Page 19 of 41

20 c in( ) c co in( ) c2 in 2c co( ) (37) An analytical olution for the optimal pitch angle wa not found from thi expreion, therefore Newton method * i once more applied to find. To enure that the optimal pitch angle doe not generate a normal vector ˆn pointing toward the Sun, bound are impoed on the optimum pitch angle, a depicted in Fig. 9 for two epoch during the year. Furthermore, by requiring to be contained in the firt two and lat two quadrant for orbit diplaced above and below the equatorial plane, repectively, a / i enured uch that the olution 2 2 SEP correpond to a minimum rather than a maximum of, a. SEP Note that Fig. 9 clearly illutrate that the out-of-plane diplaced GEO a preented in thi paper cannot be maintained throughout the year uing only a olar ail. For intance, in ummer the haded area how that the required thrut direction for a diplaced GEO diplaced above the equatorial plane (i.e. a thrut along the poitive z E axi) cannot be achieved by the olar ail. A imilar reaoning hold for a GEO diplaced below the equatorial plane in winter. Furthermore, in autumn and pring the required thrut direction for orbit diplaced both above and below the equator lie on the edge of the haded half-circle. The magnitude of the olar ail acceleration along the z E axi in that cae become equal to zero a the Sun hine edge-on to the olar ail. Winter Summer z z E E,min,min rˆ,max i obl,max x E x E i obl rˆ,min,max iobl i obl,min,max iobl i obl Fig. 9 Definition of minimum and maximum olar ail pitch angle during the year. Page 2 of 41

21 Once the optimal ail pitch and yaw angle are found, the magnitude and direction of the acceleration required from the SEP ytem can be computed. Uing Eq. (3), the pitch and yaw angle of the SEP thrut force, SEP and SEP repectively, can be computed, ee Fig. 8: SEP asep, z E aco asep atan2 a, a SEP SEP, ye SEP, xe (38) a well a the magnitude of the required SEP thrut force: Previouly it wa already tated that asep, y E ince T m a SEP (39) contant SEP yaw angle of SEP n, again with n an integer equaling either or 1.. Subtituting a, into Eq. (38) give a A noted before, the above hold for one intant in time, i.e. for a given value for m and. To find the variation of the control, acceleration, thrut magnitude and ma a a function of time over multiple orbital period, the diplaced GEO i dicretized into everal node. The node are equally ditributed over the orbit, leading to a contant time interval * t in between two conecutive node. At each node, i, the required SEP thrut magnitude can be approximated uing Eq. (39) a Ti mi asep, i. Then, auming a contant thrut magnitude during the interval t, the ma at the end of the SEP y E th i interval can be approximated through the recurrence relation: T (4) i mi 1 mi t Ip g At each node the optimum olar ail angle (and ubequently the SEP acceleration, thrut angle and thrut magnitude) can be computed. When changing from one node to the ucceive node, the change in can be computed uing Eq. (27), while the ma at the tart of the new interval can be computed uing Eq. (4). The reult after one year in a GEO diplaced 35 km along the poitive z E axi are hown in Fig. 1 and by the olid line in Fig. 11. A time interval of t.5 day i adopted, which i conidered to be mall enough to allow for a fair comparion later in the paper with the analytical analyi for SEP control in Section III. Furthermore, an initial ma of 15 kg (the maller cla of geotationary pacecraft [31]) and a pecific impule of 32 are Page 21 of 41

22 aumed. Finally, four different value for the ail lightne number are conidered,.1,.5,.1 and.2, where a value of.5 can be aumed reaonable for near-term ytem [32]. Fig. 1 Optimal olar ail (olid line) and SEP (dahed line) pitch angle for a 35 km out-of-plane diplaced GEO maintained with hybrid ail control for different value of the olar ail lightne number. a) b) Fig km out-of-plane diplaced GEO maintained with hybrid ail control for different value of the olar ail lightne number. Spacecraft ma (a) and required SEP thrut magnitude (b) auming an initial ma of 15 kg and a pecific impule of 32. Solid line indicate a year-long diplacement along the poitive z E axi. Dahed line include a eaonal tranfer between north and outh diplaced GEO. Page 22 of 41

23 Fig. 1 how that the optimal pitch angle of the olar ail decreae and the pitch angle of the SEP thruter increae for increaing value of, indicating a larger contribution from the ail to the required out-of-plane acceleration for larger value of. It alo indicate a hift in the main tak of the SEP thruter from providing the out-of-plane acceleration to compenating the in-plane component of the ail acceleration. Furthermore, ome dicontinuitie can be oberved in the profile of the SEP pitch angle for the larget value of. Thi large value for caue the component of the olar ail acceleration along the poitive z E axi to become larger than the required out-of-plane acceleration. Thi require the SEP thruter to thrut along the negative z E axi to counterbalance the acce out-of-plane acceleration, hence the witch in the SEP pitch angle from /2 to /2. In general, the larger the value for the lower the demand on the SEP ytem, which i directly tranlated into a larger final ma after 1 year in orbit when uing hybrid ail control intead of SEP control, ee Fig. 11a. Already a olar ail with =.1 provide a gain in propellant ma of 29 kg. Further increaing reult in aving of 94, 13 and 161 kg for =.5,.1 and.2, repectively. Finally, conidering the required thrut magnitude in Fig. 11b, another advantage of hybrid ail over pure SEP become evident. While the thrut level required for a 15 kg pacecraft with SEP control i larger than currently achievable thrut level of.2 N for thi pacecraft ize (e.g. EADS/Atrium RIT-XT), thrut level maller than.2 N throughout the year can be oberved for.1 and.2. Even for.5 the thrut level remain well under.2 N during winter, but it i higher during ummer. Thi performance can be improved by tranferring the pacecraft from a GEO diplaced above the equatorial plane (north) in winter to an orbit diplaced below the equatorial plane (outh) in ummer. Then, the performance of the ail i no longer limited by the obliquity of the ecliptic and can perform equally well in ummer a it doe in winter above the equatorial plane. When thi o-called eaonal tranfer i introduced in the model, reult a preented by the dahed line in Fig. 11 are obtained. Note that the miion i aumed to alway tart in winter, i.e. above the equatorial plane, and that an intantaneou eaonal tranfer i conidered. A expected, maive improvement both in term of propellant conumption and required thrut level can be oberved. The ma aving mentioned before are now increaed to 39, 129, 178 and 219 kg for.1,.5,.1 and.2, repectively. At the end of thi paper, it will be hown that SEP SEP Page 23 of 41

24 tranfer from above to below the equatorial plane and vice vera are poible and come at the cot of a negligible to modet SEP propellant conumption. While the reult in Fig. 1 and Fig. 11 only hold for a miion lifetime of one year, it i intereting to invetigate whether hybrid propulion can enable out-of-plane diplaced GEO miion lating a long a current geotationary miion. Previou ection already howed that impulive and SEP control are unable to do o. Extending the miion lifetime for hybrid ail control reult in the graph of Fig. 12. Note that all reult neglect the effect of eclipe on the performance of the olar ail. For the (diplaced) GEO, eclipe occur for a hort period per day around the equinoxe. It i aumed that increaed SEP thrut can compenate for the abence of thrut from the olar ail during thee brief period. Fig. 12 include both cae of excluding and including the eaonal tranfer and how that the eaonal tranfer can ignificantly increae the miion lifetime from a few month for the maller value for up to a few year for the larger value for. Furthermore, comparing Fig. 12 with Fig. 6 and Fig. 7 how a dramatic improvement of the lifetime for hybrid ail control compared to both impulive and SEP control. For mall diplacement ditance even lifetime of 1-15 year come into reach and the lifetime for the larger diplacement become reaonable. Again, comparing the lifetime for a ma fraction of.5 and a pecific impule of 32 increae the lifetime for a 35 km out-of-plane diplaced orbit from 3.7 year for SEP control to year (depending on the value choen for ) when the eaonal tranfer i not included and to year when the tranfer i included. Similarly, the lifetime for a 15 km out-of-plane diplaced orbit are increaed from.9 year to year (excluding tranfer) and to year (including tranfer). B. Comparion with in-plane diplaced GEO Although the analye performed in Section II and III howed that the out-of-plane diplaced GEO outperform the in-plane diplaced GEO for the ue of impulive and SEP control, it i till worthwhile to invetigate the performance of the in-plane diplaced GEO for the ue of hybrid ail. The reaon for thi i the fact that, depite the larger required acceleration to maintain the in-plane diplaced GEO, the direction of thi acceleration i much more favorable a it i approximately along the Sun-ail line in part of the orbit. Page 24 of 41

25 h = 35 km h = 75 km h = 15 km Fig. 12 Out-of-plane diplaced GEO maintained with hybrid ail control: miion time L (a maximum of 15 year i conidered) a a function of the pecific impule I p and the ma fraction mf / m, for different value of the diplacement ditance h and the ail lightne number. The olid urface exclude a eaonal tranfer between north and outh diplaced GEO. The tranparent urface include thi tranfer. To invetigate the performance of hybrid ail for the in-plane diplaced GEO, the minimization problem in Eq. 2 2 (31) need to be olved with * co 2 2 ax E GEO E, E ay GEO * in E and a, where E i the angular poition in the diplaced GEO, meaured from the poitive x E axi in counterclockwie direction. Applying the ame approach a ued in Section IV.A to olve for the optimum olar ail pitch and yaw angle would required a ytem of nonlinear equation to be olved uing the Newton method rather than the ingle expreion in Eq. (37). z E Page 25 of 41

26 Therefore, the minimization problem i olved uing a equential quadratic programming (SQP) method implemented in the MATLAB function fmincon [33]. Thi function allow to define the bound for the ail pitch angle a hown in Fig. 9 and include a contraint to enure ( nr ˆˆ ). A for the out-of-plane cae, the diplaced GEO i dicretized into node, again with a time interval of t.5 day, and at each node the minimization problem of Eq. (31) i olved. The reult for a 35 km diplaced orbit are provided in Fig. 13 and Fig. 14. Fig. 13 clearly illutrate the influence of the changing direction of the Sun-ail line during the year and the ail attitude contraint that prevent the ail from generating an acceleration in the direction of the Sun. The latter require the ail to be turned 18 every orbit and almot intantaneouly during the equinoxe. However, a expected, during part of the in-plane diplaced orbit (around E ) the ail normal i aligned with the required, radial acceleration, which ignificantly lower the demand on the SEP thruter, ee Fig. 14. Thi figure provide the acceleration required by the SEP thruter for both in-plane (olid line) and out-of-plane (dahed line) diplaced GEO for different value of the ail lightne number and during the oltice (Fig. 14a) and the equinoxe (Fig. 14b). The favourable Sun-ail line and required radial acceleration even caue the SEP acceleration for the in-plane cae to be lower than for the out-of-plane cae during the equinoxe and for.1. However, during the remainder of the orbit, the ail attitude contraint retrict the ail to uch extent that the SEP thruter ha to provide the greater part of the required acceleration, cauing the out-of-plane cae to outperform the in-plane cae alo for the ue of hybrid ail. a) b) Fig. 13 Solar ail normal vector (.1) for a 35 km in-plane diplaced GEO during the oltice (a) and equinoxe (b). Page 26 of 41

27 a) b) Fig. 14 Required SEP acceleration for a 35 km in-plane (olid line) and out-of-plane (dahed line) diplaced GEO for different value of the ail lightne number and during the oltice (a) and the equinoxe (b). The haded area illutrate the contribution from the olar ail for.1. Since all type of propulion conidered how a much better performance for the out-of-plane diplaced GEO than for the in-plane diplaced GEO, the remainder of thi paper, i.e. the ma budget analyi and the tranfer trajectorie will focu olely on the out-of-plane diplaced GEO. V. Ma budget The reult in Fig. 6, Fig. 7 and Fig. 12 provide the performance of impulive, SEP and hybrid ail control for an out-of-plane diplaced GEO in term of propellant conumption. However, the goal of the miion i to maximize the lifetime of a pacecraft carrying a given payload. It hould therefore be invetigated whether the ma fraction and pecific impule of Fig. 6, Fig. 7 and Fig. 12 allow for any payload ma to be left at the lifetime hown in thoe figure. For thi, the pacecraft ma budget i invetigated. However, due to it poor performance, impulive control i dicarded a a viable option to maintain the out-of-plane diplaced GEO and thi ection will therefore only conider the ma budget for a hybrid ail and SEP propelled pacecraft. The correponding ma budget are baed on what i propoed in Ref. [34]: m mprop mtank msep mp mgimbal m mpay (41) Page 27 of 41

28 The initial ma i broken down into even element. Firt, a propellant ma, m prop, that follow from the initial and final pacecraft ma (ee Eq. (19) and Eq. (4)), where the final ma i obtained after a certain lifetime L. Then, the ma of the tank required to tore the propellant, m.1m tank prop [35], and the ma of the SEP thruter, which i a function of the maximum power required by the SEP ubytem, of the maximum thrut required during the miion, T max : P SEP,max, which on it own i a function P m k P SEP SEP SEP,max SEP,max T I g 2 max p SEP (42) with ksep.2 kg/w [29] the pecific performance of the SEP thruter and SEP.7 [36] it efficiency. Subequently, in the cae of SEP control a olar array with ma m k P i aumed to provide electrical P SA SEP,max energy to the SEP ytem with ksa 1 45 kg/w the pecific performance of the olar array [29]. In cae of hybrid ail control, it i aumed that part of the ail i covered with thin film olar cell for thi purpoe. The required area covered with olar cell can be computed from the maximum power required by the SEP ytem: A TF P SEP,max co SEP,max (43) WTF The efficiency of the thin film i et to a conervative value of.5 and SEP,max repreent the angle TF between the Sun-ail line and the olar ail normal vector when T T max. From Eq. (43) the ma of the thin film m A can be computed with TF P TF TF 1 g/m 2 [37]. Note that the influence of the thin film olar cell on the performance of the ail i neglected in thi paper. Then, the ma of a gimbal, m.3m [35], i taken into account to enure that the olar ail and SEP thruter can teer independently of one another. Finally, the ma of the ail can be computed through m A with the ma per unit area of the olar ail and the area of the ail, given through: gimbal SEP A, m A A * TF (44) Page 28 of 41

29 For the ail loading a value of 5 g/m 2 i aumed, which i optimitic, but conidered reaonable for future olar ail miion a hown in recent tudie [38-39]. Clearly, for an SEP controlled pacecraft, both are et to zero. m gimbal and m For a given miion lifetime and for a particular pecific impule, the only unknown for computing the payload ma are the initial ma and the maximum SEP thrut required during the miion, T max, which are related ince the initial ma i bounded by T max. For SEP control, thi maximum thrut occur at t t cauing Tmax T. With the required acceleration to maintain the out-of-plane diplaced GEO given for a particular diplacement ditance, the maximum initial ma can be computed through m,max T a. However, for hybrid ail control, the maximum thrut doe not necearily occur at t t, but can alo occur in autumn (when the eaonal tranfer i taken into account, a i done in thi ection) a hown in Fig. 11b. The reulting maximum initial mae for both SEP and hybrid ail control are provided in Fig. 15 a a function of the maximum thrut magnitude and for each of the diplacement ditance ued o far and for different ail lightne number. h 15 km h 75 km h 35 km Fig. 15 Maximum thrut magnitude T max a a function of the initial ma m for different value of the out-ofplane diplacement ditance h and the ail lightne number and for Ip 32. The figure how that for SEP control and a maximum thrut magnitude of Tmax.2 N (ee Section IV.A), maximum initial mae of 174, 51 and 251 kg are poible for diplacement ditance of 35, 75 and 15 km, repectively. Thee initial mae increae by a factor 1.5 to 2.7 for hybrid ail control, depending on the ail Page 29 of 41

30 lightne number and the diplacement ditance. Thee higher initial mae how another major advantage of hybrid ail control over SEP control in addition to the propellant ma aving hown in Fig. 7 and Fig. 12. Uing thee initial mae and a pecific impule of 32, the payload mae and lifetime a depicted in Fig. 16 can be obtained. Fig. 16 how that in almot all cae hybrid ail control outperform SEP control. The only exception occur for the larget diplacement conidered in combination with the larget value for the ail lightne number,.2. Fig. 16 furthermore how that only hybrid ail control allow lifetime equal to the lifetime of current geotationary pacecraft of 1-15 year, while till enabling a coniderable payload to be taken onboard. For example, for a 35 km out-of-plane diplaced orbit, a ail lightne number of.1 and an initial ma of 2193 kg, payload mae of 487 kg and 255 kg can be maintained in the diplaced GEO for 1 and 15 year, repectively. a) b) h = 35 km h = 15 km Fig. 16 Payload ma m pay a a function of the miion lifetime L for a 35 km (a) and 15 km (b) out-of-plane diplaced GEO, for different ail lightne number and for I p = 32. The reult in Fig. 16 furthermore ugget that an optimum lightne number exit. For intance, for the 35 km diplaced GEO in Fig. 16a the reult for.1 outperform the reult for both maller and larger value for. Some detail on thi optimum lightne number are provided in Fig. 17. Thi figure how the increae in ail ma and the gain in initial, propellant, tank and power ource ma that are achieved by increaing the value for for a 35 km out-of-plane diplaced GEO. The difference between the two line i thu the net increae in payload ma. The figure clearly how that for increaing beyond a certain value, the gain in initial, propellant, tank and power Page 3 of 41

31 ource ma no longer outweigh the required increaed ail ma and increaing even further would only reult in a net decreae of the payload ma. The figure furthermore how that the optimum value for depend on the miion lifetime, which i introduced through the dependency of the ail lightne number. Note that and are therefore of no influence on the graph in Fig. 17. m prop and m tank on both the miion lifetime and m SEP and m gimbal are independent of the lightne number and miion lifetime Fig. 17 Increae in ail ma and gain in initial, propellant, tank and power ource ma due to increae in ail lightne number for a 35 km out-of-plane diplaced GEO. Note that the analyi in thi ection aume the ue of one SEP thruter. However, multiple SEP thruter could be clutered to provide a larger maximum thrut and with that a larger initial ma. Inpecting the eparate ma component in Eq. (41) how that all component cale linearly with the maximum thrut, including the payload ma. Therefore, by clutering for intance three SEP thruter to obtain a maximum thrut of.6 N, the previouly mentioned reult for a 35 km out-of-plane diplaced orbit and a ail lightne number of.1 can be increaed to an initial ma of 6579 kg and payload mae of 1461 kg and 765 kg to be maintained in the diplaced GEO for 1 and 15 year, repectively. Although the performance for a 35 km out-of-plane diplaced orbit i highly promiing, the performance of the higher diplaced orbit i not, ee Fig. 16b. The lifetime decreae dratically to approximately.5 year. Depite thi hort lifetime an intereting application exit for the 15 km out-of-plane diplaced GEO, namely to provide temporary diplacement. Then, the diplaced GEO i only maintained for a relatively hort period of time to provide ervice when needed and i tranferred into a Keplerian parking orbit when inoperative to ave propellant ma. For Page 31 of 41