Stability Analysis of Orbital Motions around Uniformly Rotating Irregular Asteroids

Transcription

1 Stability Analysis of Orbital Motions around Uniformly Rotating Irregular Asteroids By Xiyun HoU, ) Daniel J. SCHEERES, ) Xiaosheng XIN, ) Jinglang FENG, ) Jingshi TANG, ) Lin LIU, ) ) Shool of Astronomy and Spae Siene, Nanjing University, Nanjing, China ) Department of Aerospae Engineering Sienes, The University of Colorado at Boulder, Boulder, USA (Reeived June st, 07) Through the apah of periodi orbits, stability of the orbital motions around uniformly rotating irregular asteroids is analyzed. For the planar ase, studies show that the phase spae struture is quite different from that of the two-body blem when the OD terms are large, and stable orbital motions interior of the : resonane generally do not exist. For the three-dimensional orbits, studies show that the motions with orbit inlinations lose to the ritial values are unstable, thus are able to esape from or ollide with the asteroid following a non-planar route. Taking Eros as an example and onsidering higher order non-spherial terms, some extraordinary orbits are found, suh as orbits with orbital plane o-rotating with the asteroid, orbits are quasi-stationary in spae, and frozen orbits with argument of perigee different from 0, 90, 80, and 70 degrees. Key Words: asteroid, periodi orbit, resonane, stability, ritial orbit inlination. Introdution Asteroids are remnants of the early solar system and are targets of many observation/searh grams from both the ground and the spae. With the suess of several past spae missions, there is a growing interest on sending new bes to these primitive objets, suh as the two on-going missions Hayabusa- ) and OSIRIS-Rex ), and also some posed or planned missions. Different major bodies in the solar system, asteroids are usually muh smaller rregularly-shaped. This makes their gravity muh weaker rregular ompared with those of the major bodies, whih ause diffiulties to studying the be s orbital motion, for two blems: () the traditional analytial satellite theory based on perturbation theory onverges slowly or fail due to the relatively large non-spherial perturbations. () the relatively small gravity makes other perturbative fores suh as the solar radiation pressure muh larger. This paper fouses on the first blem. Through the apah of periodi orbits in the asteroid s body-fixed frame (whih an also be viewed as resonane orbits in the inertial frame), we qualitatively desribe how the phase spae struture and the stability of the orbital motions around asteroids are affeted by non-spherial terms. For the planar orbital motions and highly irregular asteroids, we find that the phase spae struture is quite different from that of the two-body blem, and generally no stable motions interior of the : resonane an be found. For the three-dimensional motion, we find that orbits with orbit inlinations lose to the ritial values are unstable. To make the results of the urrent paper more general, most of the work is arried out in the OD gravity field. However, results in the urrent work an be used as a step stone to the results in the gravity field of speifi asteroids. At the end of the work, taking the asteroid Eros and an example, and also onsidering a gravity with higher order non-spherial terms, some interesting periodi orbits are reported. This onferene paper is a brief summarize of the work presented in Ref 3) and 4). Readers who are interested in the results an find more details in these two referenes.. Equations of Motion In the body-fixed frame of a uniformly rotating asteroid, the orbit of a small body follows,,0 T T V r na y x na x, y,0 r where n a is the rotation speed of the asteroid, and V is the minus of the asteroid s potential 5). Trunated at the seond order, V takes the form of J 3 z J V 3 5 x y r r r r To attribute physial meaning to the OD terms and, we use an ellipsoid shape model with three semi-axes as a b and a onstant density of.5g/m 3 for the asteroid. We have C a b C a b 0, 5Re 0Re Instead of using a value representing the size of the asteroid for the referene radius Re, we use the height of the synhronous orbit 3 e syn a J () () J (3) R r GM (4) whih is obtained by assuming the asteroid as a partile. Eq. () admits an integral of the form v C (5)

2 where C is alled as the Jaobi onstant. Eq. () also admits four equilibrium points outside of the asteroid, two of whih lie at the long axis and the other two lie at the short axis. Viewing the equilibrium points in the inertial frame, they are atually orbits that are in : resonane with the asteroid s rotation. One remark is, only the OD gravity of the asteroid is onsidered, although the ellipsoid shape model for the asteroid is taken. 3. Stability of the Planar Motion 3.. Poinarésetion For the planar ase, Poinarémaps serve as a useful tool to separate stable orbits from unstable ones. Fig. shows the Poinarémaps for the Jaobi onstant of the equilibrium point at the asteroid s long axis. The asteroid has a shape parameter as a:b: = km: 0.65km: 0.4km. The rotation period is 36h, 30h, 4h and 0h. outside of the : resonane. In the unperturbed two-body blem, the relation between these periodi families (we use the terminology genealogy ) is shown in Fig.. The ordinate is the Jaobi onstant and the absissa is the period. Fig.. Genealogy between the three periodi families on the T-C plane for the unperturbed two-body blem. Family I are II are denoted as solid lines and Family III is denoted as dashed lines. In the presene of the OD terms, the genealogy is distorted by perturbations, as shown by Fig. 3 and Fig. 4. Fig. 3. Genealogy between periodi families for an asteroid with a:b: = km: 0.9km: 0.8km and a rotation period of h. Fig.. Poinarémaps for a same asteroid but with different rotation periods. One obvious feature of Fig. is that there are many resonane islands in Fig., and these islands gradually disappear when the asteroid rotates faster. The enters of the resonane islands are atually periodi orbits in the asteroid s body-fixed frame. The blem of the Poinarémaps suh as the one in Fig. is that we have to fix one Jaobi onstant to generate it. If the Jaobi onstant hanges, we may have a ompletely different map. The reason is that we peer the dynamis of the system from different slies of the energy but not globally. On the ontrary, we are able to globally desribe how the resonane orbits are organized in the phase spae, via the apah of periodi orbits in the asteroid s body-fixed frame, even when the non-spherial perturbations are very large. 3.. Families of periodi orbits In this study, there are three kinds of periodi orbits. Two have member with near-irular shapes in both the asteroid s body-fixed frame and the inertial frame. They are denoted as Family I and II in this study. The other one has members of eentri orbits in the inertial frame, s denoted as Family III. Family I is interior of the : resonane and Family II is Fig. 4. Genealogy between periodi families for an asteroid with a:b: = km: 0.6km: 0.4km and a rotation period of h. Comparing Fig. with Fig. 3-4 shows that eah resonane splits into two branhes, but the genealogy between periodi families generally remain unhanged although distorted from that of the BP. This means it s still per to view the orbital motions lose to the asteroid as perturbed Keplerian orbits. However, when the asteroid is even more elongated, the genealogy of periodi families is ompletely different from that of the BP. Fig. 5 shows only two periodi families for the ase of a:b: = km: 0.3km: 0.km. Fig. 5. Genealogy between periodi families for an asteroid with a:b: = km: 0.6km: 0.4km and a rotation period of h.

3 For larity, many other periodi families that are also in the same region of the T-C plane are not shown in Fig. 5. Differene between Fig. 5 and Fig. -4 indiates that maybe it is no longer valid to view the orbital motions as perturbed Keplerian orbits anymore Stability analysis Thik lines in Fig. 3 and Fig. 4 indiate the stable orbits in the families. With the inrease of the OD terms, stable region quikly shrinks on the T-C plane. For example, in Fig. 4, exept some stable : resonane orbits, generally no stable orbits exist in eah resonane. This indiates that the stable resonane islands in Fig. no longer exists, no matter what the Jaobi onstant is. A qualitative piture is given in Fig. 6. The gray ring in Fig. 6 indiates a haoti region enveloping the : resonane. In this haoti region, generally no stable resonane orbits exist exept the : one. When the asteroid beomes more elongated (a) or rotates fasters, the width of this region inreases and may eventually touhes the surfae of the asteroid. In suh ases, no stable orbits interior of the : resonane an be found. Fig. 6. Two intuitive pitures showing the ess of gradual disappearane of stable regions interior of the : resonane. sub-resonanes. The distane between these sub-resonanes is denoted as. a M a M a M a M a M a Considering the fat that ~ 3J 5os 7 4a a 0 i e 0 we have when, whih happens when the orbit inlinations are lose to the ritial values. This auses the overlap of different sub-resonanes and the haos of this resonane, a ess as illustrated by Fig. 7. In the left frame, this orbital resonane does not overlap with nearby resonanes. As a result, even the orbital motion is haoti within this resonane, the haos is not global, i.e., the orbit does not esape. However, if this resonane also overlaps with nearby resonanes, then the motion beomes globally haoti and may eventually esape this resonane, as shown by the right frame of Fig. 7. (6) One remark is: stability of grade periodi orbits shows muh better robustness than grade periodi orbits with respet to non-spherial perturbations. 4. Stability of the Three-Dimensional Motion 4.. The seular resonane Studies show that the blem of ritial orbit inlination plays an important role in determining the stability of three -dimensional orbits. When orbits have inlinations lose to the ritial values i os 5 or i i, there is no seular hange to the argument of periapsis, i.e.,. We view the blem in this work as the seular resonane whih happens between the longitude of the asending node and the angle, i.e., 0. Sine periodi orbits are resonane orbits in the inertial frame, when the three-dimensional periodi orbits have inlinations lose to the ritial values, the orbital resonane an be taken as overlapping with the seular resonane, and this auses haos nstability of the three-dimensional motion. Similar phenomena have been observed in many blems of elestial mehanis 6-8), inluding the perturbed motions by the non- spherial gravity terms 9-0). The mehanism is qualitatively desribed as follows. Take the : resonane as an example. Atually there are a series of resonane angle for this resonane, as given by Eq. (6). The differene between these resonane angles is the oeffiients of the argument of periapsis. Denote these resonane angles as 0 Fig. 7. Two intuitive pitures showing overlap between subresonanes (left) and different resonanes (right). 4.. Families of periodi orbits In the unperturbed two-body blem, three-dimensional periodi families onnet vertial bifuration orbits of Family III with those of Family I or Family II. An intuitive piture is shown in Fig. 8. Fig. 8. An illustrative piture showing the genealogy between the three-dimensional periodi families (denoted as vertial (m-n:n) resonant family or n:(m-n) resonant family in the figure), and Family I, II, and III. 3

4 However, in the presene of the OD terms, this genealogy may be broken apart by families of eentri frozen orbits. In the perturbed ase, eah three-dimensional periodi family splits into two branhes. We only take the : resonane as example. Fig. 9 shows the genealogy of this family for three asteroids with shape parameters a:b: as km: 0.99km: 0.4km (a, b), km: 0.95km: 0.4km (, d), and km: 0.6km: 0.4km (e, f). We an see that the T-C urves of these three-dimensional periodi families are distorted from vertial lines on the T-C plane by the OD terms. Moreover, eah branh of the three-dimensional periodi family is broken apart by families of eentri frozen orbits, suh as the ones enveloped in the dashed squares in Fig. 9a. influene of the seular resonane. This orbit is haoti and exhibits large vibrations in its orbit eentriity, but this orbit does not esape. This orresponds to the ase shown by the left frame of Fig. 7. (3) In the ase of Fig. 9e and f where the J term of the asteroid is muh larger, the stable region between i ompletely disappears. In this ase, not only the sub-resonanes of the : one overlap with eah other, but also the : resonane overlaps with nearby ones and ause the global instability of the orbits. One example is shown in Fig. 0, with its positions on the T-C urve denoted as a ross in Fig. 9f. In Fig. 0, two urves are given. One is for the orbit eentriity, whih shows that the eventual fate of the orbit is to esape. The other is for the orbit inlination, whih indiates that the esape route follows an orbit inlination lose to i. Fig. 9. Genealogy of three-dimensional : resonane family in the presene of OD terms. The upper and lower three frames are for different branhes of the : resonane. From left to right, the frames orrespond to asteroids with different shapes. One remark is that the genealogy of the three-dimensional periodi families is not always broken apart by families of eentri frozen orbits. Shown in Fig. 9 is just a partiular example for the : resonane. For other resonanes, there are ases where one branh is broken apart and the other branh remains a single family, or ases where both branhes remain a single family. Moreover, the genealogy of the periodi families may hange for different ombinations of the J and J terms Stability analysis In setion 4., stability perty of spatial orbital motions is already introdued. In this setion, we take speifi examples to support the arguments in setion 4.. () In the ase of Fig. 9a and b where the J term of the asteroid is very small, the stable segment on the T-C urve with orbit inlinations between i exists. Due to the extremely small J term, the width of the sub-resonanes of the : resonane whih is diretly portional to J is small. They do not overlap with eah other, and allows stable regions far away from i exist. Fig. 0a shows the time history urve of the resonane angles for two stable orbits denoted as rosses in Fig. 9a and b. () In the ase of Fig. 9 and d where the J term of the asteroid beomes larger, starting from both ends, the stable region between i shrinks. Fig. 0b shows the time history urve of one orbit whih was stable in Fig. 9b but now beomes unstable due to the Fig. 0. Left: time history urve of two stable orbits denoted as rosses in Fig. 9a and b; Middle: time history urve of the orbit eentriity for an unstable orbit denoted as a ross in Fig. 9d; Right: time history of the orbit eentriity and the orbit inlination for an unstable orbit denoted as a ross in Fig. 9f. 5. A Case Study Above work is arried out in the OD gravity. The results an help us qualitatively desribe the global dynamis around asteroids, but generally an be diretly applied to speifi asteroids. However, as already being stated previously, results of the OD gravity an serve as good initial seeds to be ontinued to the results for speifi asteroids. In this setion, we take the asteroid Eros as an example to show some results. Details of the periodi families are omitted here. Readers an found them in Refs 3) and 4). We only report some interesting periodi orbits found here. The results are presented in the 4 th order and 4 th degree gravity of Eros. 5.. Out-of-plane stable orbits o-rotating with asteroid We find that there are stable out-of-plane periodi orbits with their orbital plane o-rotating with Eros, suh as the two ones shown in Fig. a. Both orbits are grade. For fast rotating irregular asteroids suh as Eros, these orbits are of no pratial uses beause they already ollide with the asteroid s surfae. However, for slowly rotating asteroids, these stable out-of-plane orbits are of pratial uses beause they are outside the asteroid. For example, if we artifiially set Eros s rotation period as 48h, Fig. b shows two suh example orbits. Existene of these periodi orbits is due to the seular resonane na whih happens between the preession rate of the orbital asending node and the asteroid s rotation. These orbits are interesting beause they an be used to repeatedly visit ertain regions of the asteroid. 4

5 Fig.. Left: two stable out-of-plane orbits around Eros when its rotation period takes the urrent true value of 5.7h. Right: two stable out-of-plane orbits when Eros s rotation period is set as 48h. 5.. Quasi-stationary orbits We also find periodi orbits that are not only in the resonane as the one in Fig., but also in the resonane M 0. This means that the mean orbital frequeny of the orbit is zero, whih also means that the orbit does not irle the asteroid in one period. So viewing in the inertial frame, the orbit is quasi-stationary. Fig. shows suh an example in the 4 th order and 4 th degree gravity of Eros. We notie that part of the orbits lies within the Brillouin sphere where the 4 th order and 4 th degree gravity is no longer valid. This makes the results in Fig. doubtful. Existene of suh orbits in the asteroid s real gravity still requires further work. n a Fig. A quasi-stationary orbit around the asteroid Eros 6. Conlusion This paper desribes the global dynamis around uniformly rotating irregular asteroids, in a OD gravity, through the apah of periodi orbits, in the body-fixed frame of the asteroid. For the planar ase, studies show that for extremely elongated asteroids, the phase spae struture is different from that of the BP, indiating that it is no longer to treat the orbital motion lose to the asteroid as Keplerian orbits. For highly elongated asteroids, orbital motions interior of the : resonane are generally unstable. For three-dimensional orbits, their stability is influened by the seular resonane. Near irular orbits with orbit inlinations lose to ritial values are generally unstable. Aknowledgments X.Y.H thanks the National Natural Siene Foundation of China (No 3330). Referenes ) Tsuda Y., Yoshikawa M., Abe M., et al., System design of Hayabusa --- asteroid sample return mission to 999 JU3, Ata Astronautia, 9 (03), pp ) Sheeres D.J., Hesar S.G., Tardivel S., et al., The geophysial environment of Bennu, Iarus, 76 (06), pp ) Hou X.Y., Xin X.S., Feng J., Periodi orbits around uniformly rotating asteroids, Communiations in Nonlinear Siene and Numerial Simulations, submitted 4) Feng J. and Hou X.Y., Dynamis of equilibrium points in a uniformly rotating seond order and degree gravitational field, the Astronomial Journal, submitted 5) Kaula W. M., Theory of satellite geodesy: appliations of satellites to geodesy, Waltham, Mass:Blaisdell, 966 6) Moons M., Review of the dynamis in the Kirkwood gaps, Celestial Mehanis and Dynamial Astronomy, 65 (997), pp ) Robutel P. and Gabern F., The resonant struture of Jupiter s Trojan asteroid-i. long-term stability and diffusion, Monthly Noties of the Royal Astronomial Soiety, 37 (006), pp ) Hou X.Y., Sheeres D.J. and Liu L., Dynamis of the Jupiter Trojans with Saturn s perturbation in the present onfiguration of the two planets, Celestial Mehanis and Dynamial Astronomy, 9 (04), pp ) Delhaise F. and Henrard J., The blem of ritial inlination ombined with a resonane in artifiial satellite theory, Celestial Mehanis and Dynamial Astronomy, 55 (993), pp ) Saedeleer B.D. and Henrard J., The ombined effet of J and C on the ritial inlination of a lunar orbiter, Advanes in Spae Researh, 37 (006), pp