ARKOVSK EFFECT IN GENERALIZED PHOTOGRAVITATIONAL -BODIES PROBLEM Serge V. Ershkov Institute for Time Nature Eplorations M.V. Lomonosov's Moscow State Universit Leninskie gor - Moscow 999 Russia e-mail: sergej-ershkov@ande.ru Abstract: Here is presented a generaliation of photogravitational restricted -bodies problem to the case of influence of arkovsk effect which is known as reason of additional infinitesimal acceleration of a small bodies in the space due to anisotropic re-emission of absorbed energ from the sun other stellar sources. Asteroid is supposed to move under the influence of gravitational forces from massive bodies which are rotating around their common centre of masses on Kepler s trajectories as well under the influence of pressure of light from both the primaries. Analing the ODE sstem of motion we eplore the eistense of euilibrium points for a small bod asteroid in the case when the -nd primar is non-oblate spheroid. In such a case it is proved the eistence of maimall 56 different non-planar libration points in generalied photogravitational restricted -bodies problem when we take into consideration even a small arkovsk effect. AMS Subject Classification: 7F5 7F7 Ke Words: arkovsk effect ORP-effect photogravitational restricted three bod problem stabilit euilibrium points libration points oblateness
. Introduction. The arkovsk effect is a force acting on a rotating bod in space caused b the anisotropic emission of thermal photons which carr momentum Radievskii 954. It is usuall considered in relation to meteoroids or small asteroids about cm to km in diameter as its influence is most significant for these bodies. Such a force is produced b the wa an asteroid absorbs energ from the sun and re-radiates it into space as heat b anisotropic wa. In fact there eists a disbalance of momentum when asteroid at first absorbs the light radiating from the sun but then asteroid re-radiates the heat. Such a disbalance is caused b the rotating of asteroid during period of warming as well as it is caused b the anisotropic cooling of surface & inner laers; the processes above depend on anisotropic heat transfer in the inner laers of asteroid. During thousands of ears such a disbalance forms a negligible but important additional acceleration for a small bodies so-called arkovsk effect. Thus arkovsk effect is small but ver important effect in celestial mechanics as well as in calculating of a proper orbits of asteroids & other small bodies. Besides arkovsk effect is not predictable it could be onl observed & measured b astronomical methods; the main reason is unpredictable character of the rotating of small bodies Rubincam even in the case when there is no an collision between them. If regime of the rotating of asteroid is changing we could observe a generaliation of arkovsk effect i.e. the arkovsk O'Keefe Radievskii Paddack effect or ORP effect Rubincam.
. Euations of motion. Let us consider the sstem of ordinar differential euations for photogravitational restricted -bodies problem at given initial conditions Radievskii 95; Shankaran et al.. In according with Shankaran et al. we consider three bodies of masses m₁ m₂ and m such that m₁ > m₂ and m is an infinitesimal mass. The two primaries m₁ and m₂ are sources of radiation; ₁ and ₂ are factors characteriing the radiation effects of the two primaries respectivel {₁ ₂} - ]. We assume that m₂ is an oblate spheroid. The effect of oblateness is denoted b the factor A₂. Let ri i = be the distances between the centre of mass of the bodies m₁ and m₂ and the centre of mass of bod m Shankaran et al.. Now the unit of mass is chosen so that the sum of the masses of finite bodies is eual to. We suppose that m₁ = - μ and m₂ = μ where μ is the ratio of the mass of the smaller primar to the total mass of the primaries and μ /. The unit of distance is taken as the distance between the primaries. The unit of time is chosen so that the gravitational constant is eual to Shankaran et al.. The three dimensional restricted -bodies problem with an oblate primar m₂ and both primaries radiating could be presented in barcentric rotating co-ordinate sstem b the euations of motion below Shankaran et al. ; Douskos et al. 6: n n.
4 - where - is the angular velocit of the rotating coordinate sstem and A₂ - is the oblateness coefficient. Here - where AE is the euatorial radius AP is the polar radius and R is the distance between primaries. Besides we should note that - are the distances of infinitesimal mass from the primaries. We neglect the relativistic Ponting-Robertson effect which ma be treated as a perturbation for cosmic dust or for small particles less than cm in diameter see Chernikov Chernikov 97; Kushvah et al. 7 as well as we neglect the effect of variable masses of -bodies Singh et al... r r A r r n A n 5 R AP AE A r r
. Modified euations of motion arkovsk effect. Modified euations of motion. for the generalied three dimensional restricted - bodies problem with an oblate primar m₂ both primaries radiating and the infinitesimal mass m under the influence of arkovsk effect should be presented in barcentric rotating co-ordinate sstem in the form below: n t n t. t - where t t t are the projecting of arkovsk effect acceleration t on the appropriate ais O O O. 4. Location of Euilibrium points. The location of euilibrium points for sstem. in general is given b conditions: 4. t t t. 5
6 Let us consider the case when the effect of oblateness is absent A₂ = n = see the appropriate epression: It means a significant simplifing of epression. in the sstem of eualities 4.: Besides we assume all euations 4. to be a united sstem of algebraic euations. That s wh we substitute an epression for from -rd euation above to the -nd & the -st euation:. 4.. A n
7 Moreover we obtain from the -d euation of sstem 4. that planar euilibrium points eist onl if { = = } simultaneousl. But the case = is ver rare specific condition for asteroid which has unpredictable character of the regime of rotating during a flight through the space Rubincam ; the same is obtained for the case =. Therefore we will consider onl non-planar euilibrium points. So we obtain from the -st & -d euations of sstem 4.: Hence we finall obtain the sstem of algebraic euations for meanings of { } which determine the location of euilibrium points 4.: 4.4. 4.
8 - where The last sstem 4.4 could be presented as below: - where the maimal polnomial order of euations is eual to 6 6 = 56: indeed the order of -st polnomial euation is eual to 6 in regard to variables ; the order of -nd polnomial euation is also eual to 6 in regard to. So 4.4 is the polnomial sstem of euations of 56-th order which has maimall 56 different roots we should especiall note that each of them strongl depends on various parameters { μ ₁ ₂ ; }. Such a sstem of polnomial euations could be solved onl b numerical methods in general it is valid for polnomial euation of order > 5. Besides analsing the euations of sstem 4. we should note that a case of arkovsk effect is negligible determines the eistence of uasi-planar euilibrium points in which conditions { } are valid simultaneousl.. 6 6
9 To give some estimation or numerical results we should take into consideration the negligible character of arkovsk effect { } in the last sstem of euation of 56-th order. Such a simplification let us obtain the result below see 4.: If we substitute the appropriate meanings of coordinates for triangular libration points L₄ and L₅ in 4.5 when arkovsk effect euals to ero we will obtain that all the eualities are valid in terms of generalied photogravitational restricted -bodies problem Xuetang et al. 99. Each of euations of sstem 4.5 has 7-th order in regard to variables so 4.5 is the polnomial sstem of euations of 49-th order which has maimall 49 different roots. That s wh let us make the net step for simplifing of the sstem 4.5: 4.5 4.6
Each of euations of sstem 4.6 has -d order in regard to variables so 4.6 is the polnomial sstem of euations of 6-th order which has maimall 6 different roots. Such a sstem of polnomial euations could be also solved onl b numerical methods it is valid for polnomial euation of order > 5. Let us present the solution which differ from the libration points L₄ and L₅ due to arkovsk effect as below: - where ₀ ₀ the appropriate meanings of coordinates of the triangular libration points L₄ and L₅ in generalied photogravitational restricted -bodies problem when = = = Xuetang et al. 99. So from 4.6 we obtain Δ Δ : The strongest simplifing of sstem 4.4 is possible when arkovsk effect is ero = = =. In such a case it has been proved the eistence of maimall 9 different euilibrium points {L₁ L₉} in photogravitational restricted -bodies problem Xuetang et al. 99..
5. Conclusion. It has been proved the eistence of maimall 56 different non-planar euilibrium points in generalied photogravitational restricted -bodies problem when we take into consideration even a small arkovsk effect in the case the -nd primar is non-oblate spheroid. This result is different both from classical restricted -bodies problem and generalied photogravitational restricted -bodies problem. Stabilit of such a points is an open problem in celestial mechanics for the case of non-ero arkovsk effect Radievskii 95. This model ma be applied to eamine the dnamic behaviour of small rotating objects such as meteoroids or small asteroids about cm to km in diameter. For the meteoroids less than cm in diameter we should additionall take into consideration the relativistic Ponting-Robertson effect which ma be treated as a perturbation for cosmic dust see Chernikov Chernikov 97; Kushvah et al. 7. arkovsk effect does not make an significant influence in regard to the meteoroids more than km in diameter Radievskii 954. Acknowledgements I am thankful to CNews Russia project Science & Technolog Forum branch Gravitation - for valuable discussions in preparing this manuscript. Especiall I am thankful to Dr. P.Fedotov Col. L.Vladimirov Dr. A.Kulikov for valuable suggestions in preliminar discussions of this manuscript. References:
Chernikov A 97. The Photogravitational Restricted Three-Bod Problem. Soviet Astronom Vol. 4 p.76. Douskos CN & Markellos VV 6. Out-of-plane euilibrium points in the restricted three bod problem with oblateness. A&A Vol. 446 pp.57-6. Kushvah BS Sharma JP and Ishwar B 7. Nonlinear stabilit in the generalised photogravitational restricted three bod problem with Ponting-Robertson drag. Astrophs Space Sci Vol. No. -4 pp. 79-9. Radievskii VV 95. The restricted problem of three bodies taking account of light pressure. Akad. Nauk. USSR AstronJournal Vol. 7 p. 5. Radievskii VV 954. A mechanism for the disintegration of asteroids and meteorites. Doklad Akademii Nauk SSSR 97: 49 5. Rubincam David P. Radiative spin-up and spin-down of small asteroids Icarus 48. Shankaran Sharma JP and Ishwar B. Out-of-plane euilibrium points and stabilit in the generalised photogravitational restricted three bod problem. Astrophs Space Sci Vol. No. pp. 5-9. Singh J Leke O. Stabilit of the photogravitational restricted three-bod problem with variable masses. Astrophs Space Sci 6: 5 4. Xuetang Zh Lihong 99. Photogravitationall Restricted Three-Bod Problem and Coplanar Libration Point. Chinese Phs. Lett. 99 Vol. pp.6-64.