Satellite s Orbital Dynamic and Stable Regions near Equilibrium Points of Asteroid

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1 International Conference on Computer and Automation Engineering (ICCAE ) IPCSIT vol. 44 () () IACSIT Pre, Singapore DOI:.7763/IPCSIT..V44. Satellite Orbital Dynamic and Stable Region near Equilibrium Point of Ateroid CuiHutao a, ZhangZhenjiang a, b,* and YuMeng a a Harbin Intitute of Technology, 9 XiDaZhi treet, Harbin, 8, China b Chongqing Changan Indutry Group Co. LTD, 99 KongGang Street, ChongQing, 4, China Abtract. In order to tudy the atellite moving feature and the ize of table reign around the center equilibrium point of ateroid, : reonant orbit theory are applied in thi paper. Firt the Hamilton function decribed by Delaunay variable i implified to Schubart tandard. Secondly the implified mathematical model i ued to dicu the form of orbital motion, vibration period of atellite which locate orbit the ateroid equilibrium point and the relationhip between each other. And then Ateroid Veta i taken a a imulation example to verify the correctne of thee concluion above. Finally, exploiting the propertie of energy integral of Schubart tandard, the formula of deducing the radiu of table region around the center point i given, which i alo applied to analyze the propertie of orbital motion both inide and outide of the table region. The reearch in thi article not only can help u better undertand the equilibrium point of ateroid and the dynamical environment in the vicinity of which but alo can it upply theoretical bai in deigning the orbit around the equilibrium point around the ateroid. Keyword: ateroid, equilibrium point, orbital reonance, table region;. Introduction There exit four equilibrium point around an ateroid [], thee four point and the dynamical environment around them are quite imilar to the Lagrange point in three-body ytem. D. J. Scheere utilized ellipoid integral function to find out the computational method of determining the poition of equilibrium point around the homogeneou triaxial ellipoid and the tability condition of equilibrium point, according to the tability condition the equilibrium point can be claified into two claification: Two untable point which locate on both ide of emi-major axi of ellipoid are named a Saddle Point; The other table couple which locate on both ide of emi-minor axi are named a Center Point. Need to mention that the definition of Saddle Point and Center Point here are different from thoe defined in Dynamic Sytem, detail can be checked in []. Saddle Point and Center Point in thi paper obey the definition given by D. J. Scheere. Due to trixial ellipoid cannot decently reflect the propertie of ateroid gravity field and the computation of ellipoid integral function i too complicated, the form of which i alo obcure to help better undertand the parameter of ellipoid hape and the relationhip between the poition and tability of center point, thi method han t been widely applied, thu in order to olve the problem above, W.hu tudied the related problem of equilibrium point in random econd degree econd order gravity field[,3] and deduced the approximate calculation formula of poition of equilibrium point which are expreed by pherical harmonic coefficient, he alo gave the tability criterion of equilibrium point. Although W. hu work olved the problem exited in D. J. Scheere reearch, it till not enough of only knowing the poition and tability of equilibrium point for an ateroid. Since poitioning the atellite at the ateroid equilibrium point cannot be realized in engineering, atellite motion around the table equilibrium point and the ize of table region become the principal problem which need to figure out currently. Changyin Zhao employed : orbit reonance theory in hi reearch of table region of triangular libration point (Lagrange point L 4, L which are imilar to the table equilibrium point of ateroid)of planet in olar * Correponding author. Tel.: addre: river8@gmail.com. 3

2 ytem[4,]. Due to the dynamical environment around the table center point i imilar to the dynamical environment around Lagrange point L 4 L of planet, : orbit reonance i adopted in thi paper to tudy the related problem about atellite orbiting around the equilibrium point of ateroid. The problem can be concluded a two repect: Firt, the form of movement and moving characteritic of atellite whoe initial poition ha a contant deviation from the center equilibrium point. Second, the problem about table region around table region around table center equilibrium point, that i to ay to control the initial deviation in what range can guarantee the atellite moving around the equilibrium point intead of moving far away.. The Poition of Equilibrium Point and It Stability In the body-fixed frame of ateroid, the approximate computational formula of poition of equilibrium point which utilize normalized unit [6] can be expreed a 3C.C xeq ± r + r 3C +.C yeq ± r r /3 Where r = ω T, which i the ynchronou orbit radiu, C and C are the pherical harmoniou coefficient of gravity field accord to the oblatene and ellipicity of center gravitational body. Thee four equilibrium point are addle point ( ± x eq,) and center point (, ± y eq ). Equation () i the approximate computational formula of the table poition in econd degree econd order gravity field. Taking the reult a initial value, by employing numerical method can the equilibrium poition in random degree and order gravity filed be obtained, Fig. i the ditribution map of the poition of equilibrium point in ixteenth order gravity field of ateroid Ero 433. For the addle point, it quite untable for the atellite moving in the vicinity of it. Only minor deviation can caue the atellite orbit diverge in the form of index and finally the atellite will either ecape from the ateroid or collide with the ateroid. Fig. i the atellite orbital condition, where the atellite initial poition i around the ateroid Ero 433 addle point. The time of orbital recurrence i.947 day. () Center Point (,4.89) Y/km Saddle Point (-.4, ) Saddle Point (8.68, ) Center Point (,-4.3) X/km Fig.. Equilibrium point of Ero 433 Fig..Orbit with Initial Poition in Saddle Point of Ero433 From Fig. we can ee that the ynchronou orbit didn t come into being, on the other hand, the atellite collided with the ateroid due to the perturbation within one day, which illutrate that the ateroid addle point are untable, if the pherical harmoniou coefficient of the ateroid atifie equation () r + C 6C > () Table. The Stability of Equilibrium Point in Celetial Bodie Center Body Earth Veta Ero433 3 μ ( km / ) C C ω ( T / ) r ( km ) r + C 6C.8 >.9 >. < Stability Stable Stable Untable 4

3 Then thee two center point are table [3], alo it mean that the atellite whoe initial poition locate at thee two point will remain the poition at ret relative to ateroid, even if there exit ome deviation, the orbit of atellite cannot diverge. Table lit the phyical parameter like the ma, angular pin rate, oblatene and ellipicity of ateroid Veta and Ero 433, alo the tability criterion of center point in the form of equation () i given. Table alo lit the related data of center point of earth in comparion. 3. The Orbit Motion around Stable Center Point The problem about poition of equilibrium point and tability of ateroid have been analyzed above. For the addle point and untable center point, the trongly intability of which can be utilized to deign ome ecape orbit or capture orbit. For the table center point, we can utilize it tability to arrange retranlator atellite or ynchronou atellite which can be aigned a long-time obervation miion of ateroid. In the following of thi ection the orbit motion around table center point of ateroid i analyzed, due to the motion in the Z direction can reolve variable and i relatively eay to analyze[7], thi paper only analyze the atellite orbit motion in the ateroid equatorial plane. Employing Delaunay variable, omitting the perturbation in higher than econd-order, after the tranformation, the hort-period perturbation can be eliminated, then the Hamilton function can be obtained a [4,] C H = + ω T L 3 3 ( 3co i ) L 4LG (3) 3C + 6 ( + coi) col 4L L i a pair of conjugate variable, the definition of which i a follow L = a l = ( Ω+ ω + M) θ, θ = S + ωt ( t t ) Where t i epoch time, S i the local idereal time of x axi in body fixed frame of ateroid. And π β = L, α = l C B β 4 β ( e ) 3C 6 ( + coi) 4β 3C A( β ) = 6 ( + coi) 4β ( β) = + ωt β ( 3co i ) 6 3 Then Hamilton function can be written a tandard Schubart model, which correpond to a ideal reonance ytem with ingle DOF ( β, α ), the reonance variable are ( β, α ), α i the angle variable, β i the reonance variable which conjugate of α. The correponding Hamilton function i H( β, α) = B( β) + A( β)in α (6) Where the econd term of the right ide of equation i the major reonance term, the relationhip below i alo tenable. A( β) >, A( β)/ B( β) = O( ε ) (7) Where the mall parameter ε indicate the magnitude of perturbation in Hamilton function, from the property of Hamilton function we know that the motion equation i H β= = P ( β, α) = A ( β) inα α H α = = Q ( β, α ) = B ( β) A ( β) in α β (4) () (8)

4 Which ha energy integral H (, ) Where C i the energy contant, differentiate the econd equation in (8) by time α Introduce equation (8) into equation () can we get Around center equilibrium point (, ± y eq ) βα = C (9) = ( B A in α) β (A in α) α α = ( AB + A B )in α + 4 AA + ( A ) in αin α () B, B = O A= O( ε), A = O( ε), A = O( ε) Then introduce equation (), expand inx in Taylor erie at equilibrium point, we can obtain 3 α = ( 4 AB ) α + O( α ) Around center equilibrium point 4AB >, for ω = 4AB and omit the high-order term, the motion equation of original ytem can be implified a imple harmonic motion equation The olution of (4) i α + ωα= () () () (3) (4) α = A co( ω t+ α ) () Where α i the initial phae of α when t =, AZ i the amplitude, ω = 4AB i vibration angular frequency. Equation () i the vibrating condition around center equilibrium point (, ± y eq ), which i the orbital reonance property in deep vibration region. From the relationhip between period and angular frequency we can get π π T = = ω 4AB (6) Table. The Stability of Equilibrium Point in Celetial Bodie Swing peroid Swing period The analyi above indicate that when the pacecraft i moving in the vicinity of center equilibrium point, it mode of motion ha the imilar property with the mathematical pendulum a their periodicity, the reonance period i almot a contant which i nearly irrelevant to the amplitude, pecific value can be calculated by equation (6). Table give the reonance amplitude of ateroid Veta ynchronou orbit atellite and the correponding period, form table we know that in deep vibration region( amplitude i below degree), the amplitude increae while period only change a little. Thu when the amplitude keep increaing till it reache light vibration region, the increaing of amplitude will caue the period a vat change. Fig. 3. Reonance Orbit in Deep Vibration Area of Veta 6

5 Fig. 3 i the chematic illutration of Veta reonance orbit when amplitude i. degree, from Fig. 3 we know that in the body fixed frame of ateroid, atellite move in the form of reciprocating vibration around center equilibrium point, the orbital hape i like a flat hoof. Fig. 4 i the reonance orbit when amplitude i 77. degree. It i een from thi that, when the amplitude i mall, atellite can till be reonated teady around center equilibrium point in light vibration region, when the amplitude i overized, the vibration become untable, but the atellite till remain within the vicinity of equilibrium point. Fig. 4. Reonance Orbit in Light Vibration Area of Veta 4. Stable Region of Orbit around the Equilibrium Point Fig.4 indicate when the amplitude i overized, the tability of reonance orbit in light vibration region become weakened. Thi ection i going to deal with the problem of what extent the amplitude increae to will the reonance orbit become untable. Equation (6) doen t obviouly contain t, thu β can be treated a the function of α, by differentiating equation (9) of α, we can get dβ Ain α P( β, α) dα = B A in α = Q( β, α) The point which equal zero in (7) i the tationary point of original function, if β acquire maximum then Ainα B A in α = i for ure, but at the non-equilibrium point mut have B + A in α, hence the maximum of β mut correpond to Ainα =, that i to ay α =. Form equation (6) and (9) we know on the boundary line of reonance region C* = B( β) + A( β) = B( β*) (8) Where β * i the maximum of β on the boundary line, C * i the correponding energy contant, β i the value of β which correpond to addle point ( ± x eq,), due to the difference between β * and β i very mall, the B( β *) can be expanded a β 4, the expreion i B( β*) B( β ) B ( β ) β B ( β )( β) (7) = + Δ + Δ + (9) Where Δ β = β * β, i the table reonance region, introduce equation (9) into (8) B( β ) + A( β ) = B( β*) = B β + B β Δ β + B β Δ β + o Δβ 3 ( ) ( ) ( )( ) ( ) We can ee from the exitence condition of addle point Q( β, α ) = that which can be introduced into () and then Becaue of B O( ε), A O( ε), A O( ε) order term we can obtain the reduced form () B ( β ) + A( β ) = () 3 B ( β)( Δβ) A ( β) Δβ A( β) + o( Δ β ) = () = = =, only β O ( ε ) Δ = can equation () be atified, omitting the high 7

6 B ( β)( Δβ) A( β) = (3) Therefore the approximate formula of table reonance region can be expreed a Δ β = When calculating the reonance period T and table reonance region Δ β, B i very needed to calculate, thu the approximate calculation of B become important, differentiate the econd equation in () twice can we get 4 B = 3L + O( C ) () Then we can finally obtain the reonance period and table reonance region around the center equilibrium point by equation () () and (6), the final reult of which i A B (4) + ini Δ β = C (6) L Need to mention here i the definition of calculation reult L from Delaunay variable i L = a if uing tandard unit, now we tranform it into Careian vector variable and introduce () into (6), then Δ y = C r r.c + 3C Equation (7) i the formula of table reonance region in the equatorial plane of ateroid, Figure i the condition of reonance orbit of atellite locating on the edge of reonance region, from Fig. we can ee that after a reonance period the atellite i unable to return to it original poition, which indicate the reonance orbit of which i already untable, only ome mall perturbation or long-time orbital movement can caue the atellite move far away from the center point. (7). Concluion Fig.. Orbit in Stable Region Edge of Veta Thi paper tudied the propertie of atellite orbital motion around the table center point of ateroid, which can be recognized a the inheriting and developing of what predeceor did. : orbit reonance theory i adopted to deduce the atellite orbital motion around center equilibrium point (in the deep reonance region), which i in a reonance condition, the reonance tate of angular motion i quite imilar to the property of mathematical pendulum. The reonance period i almot a contant which i irrelevant to amplitude, the value of period can be calculated approximately by equation (6). The approximate width of table reonance region can be obtained by equation (7). In thi region the atellite orbit i table which mean neither can atellite ecape from ateroid nor collide with ateroid. If atellite exceed the table region, it orbit will become untable, ome mall perturbation or long-time orbital motion may caue the atellite ecape or colliion. The reearch of thi paper can not only help u better undertand the equilibrium point of atellite and the dynamical environment around it but alo of great upport in providing the theoretical bai when deigning the orbit around equilibrium point of ateroid. 8

7 6. Reference [] Scheere D J. Dynamic about uniformly rotating triaxial ellipoid: Application to ateroid, Icaru, 994,, pp []Hu W, Scheere D J. Periodic orbit in rotating nd degree and order gravity field. Ninth International Space Conference of Pacific-bain Societie (ISCOPS), Paadena, CA.. [3]W. Hu and D. J. Scheere. Numerical Determination of Stability Region for Orbital Motion in Uniformly Rotating Second Degree and Order Gravity Field, Planetary and Space Science.4., pp [4]Zhao Changyin, Liu Lin. Stable region around triagle equilibrium point [J]. Atronomy :Vol 34 No. []Zhao Changyin, Liu Lin. Stable region around triagle equilibrium point(ii)[j]. Atronomy :Vol 3 No. [6]Bate. R, Meuller. D and White. J. Fundamental of Atrodynamic[M]. Dover Publication, INC, New York, [7]Liu Lin, Wang Xin. An orbital Dynamic of Lunar Probe[M]. BeiJing: National Defence Indutry Pre,6.9, pp