Periodic Orbits in the Photogravitational Restricted Problem When the Primaries Are Triaxial Rigid Bodies

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1 International Journal of Astronomy and Astropysics Publised Online Marc 06 in SciRes. ttp:// ttp://dx.doi.org/0.46/ijaa Periodic Orbits in te Potogravitational Restricted Problem en te Primaries Are Triaxial Rigid Bodies Preeti Jain Rajiv Aggarwal Amit Mittal Abdulla Department of Matematics A.R.S.D. College University of Deli ew Deli India Department of Matematics Sri Aurobindo College University of Deli ew Deli India Department of Matematics College of Science in Zulfi Majmaa University Riyad KSA Received December 05; accepted 6 Marc 06; publised 0 Marc 06 Copyrigt 06 by autors and Scientific Researc Publising Inc. Tis work is licensed under te Creative Commons Attribution International License (CC BY). ttp://creativecommons.org/licenses/by/4.0/ Abstract e ave studied periodic orbits generated by Lagrangian solutions of te restricted tree-body problem wen bot te primaries are triaxial rigid bodies and source of radiation pressure. e ave determined periodic orbits for different values of µ A A A A P and P ( is energy constant; μ is mass ratio of te two primaries; A A A and A are parameters of triaxial rigid bodies and P and P are radiation parameters). Tese orbits ave been determined by giving displacements along te tangent and normal at te mobile co-ordinates as defined in our papers (Mittal et al. []-[]). Tese orbits ave been drawn by using te predictor-corrector metod. e ave also studied te effect of triaxial bodies and source of radiation pressure on te periodic orbits by taking fixed value of μ. Keywords Restricted Tree-Body Problem Periodic Orbits Triaxial Rigid Body Radiation Pressure. Introduction Tis paper is te extension of our papers Mittal et al. []-[]. Carlier [4] and Plummer [5] studied te existence of two families of small periodic motions in te neigborood of Lagrangian solutions in te plane circular restricted tree-body problem wit different values of te parameter μ. Riabov [6] investigated periodic motions analytically. Szebeely [7] as described te results on te periodic motions of circular restricted tree-body problem. Deprit and Henrard [8] gave more results on periodic motions in teir paper. Markeev and Sokolsky How to cite tis paper: Jain P. Aggarwal R. Mittal A. and Abdulla (06) Periodic Orbits in te Potogravitational Restricted Problem en te Primaries Are Triaxial Rigid Bodies. International Journal of Astronomy and Astropysics 6 -. ttp://dx.doi.org/0.46/ijaa

2 [9] worked on te small periodic motions generated by Lagrangian solutions for all values of μ and for small values of for wic te conditions of olomorpic integral teorem are valid. Hadjidemetriou [0] as discussed te continuation of periodic orbits from te restricted to te general tree-body problem. Karimov and Sokolsky [] ave studied te periodic motions generated by Lagrangian solutions of te circular restricted tree-body problem by using mobile co-ordinates and by taking displacements along te tangent and te normal. Aggarwal et al. [] ave discussed te non-linear stability of te triangular libration point wen bot te primaries are radiated axe symmetric rigid bodies in te presence of tird and fort order resonance. Abouelmagd et al. [] ave studied te periodic structure of te restricted tree-body problem considering te effect of te zonal armonics J and J 4 for te more massive body. Tey sowed tat te triangular points in te restricted tree-body problem ave long or sort periodic orbits in te range 0 µ < µ c. Perdios et al. [4] ave studied te equilibrium points and related periodic motions in te restricted tree-body problem wit angular velocity and radiation effects. Jain and Aggarwal [5] ave studied te stability and existence of non-collinear libration points in restricted tree-body problem wit stokes drag effect wen smaller primary is an oblate speroid. Te celestial bodies are in general axis-symmetric bodies so its sape sould be taken into account as well. Te replacement of mass point by rigid-body is quite important because of its wide applications. Te re-entry of artificial satellite as sown te importance of periodic orbits. Tat is wy we ave tougt of studying in tis paper te periodic orbits generated by Lagrangian solutions of te restricted tree-body problem wen bot te primaries are triaxial rigid bodies and source of radiation pressure. e determine te periodic orbits by giving displacements at te mobile co-ordinates along te tangent and normal. e ave also determined family of periodic orbits by fixing μ (mass ratio of te two primaries) and canging te values of A A A A (parameters of te triaxial rigid bodies) P and P (te radiation parameters) and varying (energy constant). e ave also studied te effect of triaxial parameters and te radiation pressure on te energy constant (). Most of te autors ave not taken into account te effect of te solar radiation pressure in te motion of te tird body wereas we ave taken bot te primaries as radiating triaxial rigid bodies. Besides taking bot te primaries as triaxial rigid bodies and te source of radiation we ave used mobile-coordinates and given te displacement along te normal and te tangent to te orbit wic as wider applications in space dynamics. e ave drawn te periodic orbits by using te predictor-corrector metod wic is given in detail in our papers []-[].. Equations of Motion Following te procedure of our papers []-[] we consider tree masses m m and m wit m m and bodies wit masses m and m revolve wit angular velocity n (say) in circular orbits witout rotation about te centre of mass O. Let tere be an infinitesimal mass m wic is moving in te plane of motion of m and m and is being influenced by teir motion but not influencing tem. Let te line joining m and m be taken as X-axis and O teir centre of mass as origin. Let te line troug O and perpendicular to OX and lying in te plane of motion of m and m being Y-axis. Let us consider a synodic system of coordinates O(XYZ) initially coincident wit te inertial system O(XYZ) rotating wit te angular velocity n about Z-axis; (te Z-axis is coincident wit Z-axis). e coose unit of mass suc tat m+ m = te unit of distance AB = and unit of time is so cosen tat G =. Using te dimensionless variables we find te Lagrangian function L and te equations of motion of te infinitesimal mass in te restricted tree-body problem wen bot te primaries are radiating triaxial rigid bodies in te synodic co-ordinate system (Figure ). Equations of motion wit Lagrangian function L are given by were d L L d L L = 0 = 0 dt x x dt y y n A Ay P L = ( x + y ) + n( xy xy ) + ( x + y ) + ( µ ) r r r r A Ay P U 5 r r r r + µ ()

3 Figure. Configuration of te restricted tree-body problem wit masses m and m as radiating triaxial rigid bodies. ( x y) = m te synodic rectangular dimensionless co-ordinates of te infinitesimal mass. Here we ave assumed bot te primaries wit masses m and m are radiating triaxial rigid bodies n = mean motion = + A+ A ; A and A are te parameters of triaxial rigid bodies. It may be noted 4 4 tat n is independent of A and A and P and P (te radiation parameters) Radiation pressure due to te bigger primary P = Gravitational force due to te bigger primary Radiation pressure due to te smaller primary P = Gravitational force due to te smaller primary a b c a b A A c = = 5R 5R b a b a A = A = ( A A A A ) 5R 5R a b c = lengt of te semi axes of te triaxial body of mass m a b c = lengt of te semi axes of te triaxial body of mass m R = dimensional distance between te primaries r ( µ ) r ( µ ) x y x y = + + = + + m µ = m + m U = constant to be so cosen suc tat (energy constant) will vanis at L 4 (libration point). Te coordinates of L 4 (libration) are xl = + µ + α 4 A+ α A + γa + γ A + βp+ β P yl = + α 4 A+ α A + γa + γ A + βp+ β P

4 were α = α = α = α = 7 γ = 8 µ 7 γ = + 8 ( µ ) 5 γ = 4 µ γ = 5 4 ( µ ) and β = ( ) β = β = β = A A U = ( µ + µ ) ( µ + µ ) ( 5 5 µ + µ ) 4 4 A A ( 6µ + µ µ 4) + µ + 4µ 7µ 8 µ 8 µ + µ P+ µ P. Equations of motion can also be written as were Te Jacobi integral is n = x + y x ny = x y ( ) ( µ ) ( ) ( ) y + nx = () A Ay P r r r r A Ay P + µ + + U r r r r n ( ) ( ) ( µ ) C = x + y x + y. ormal and Tangent ariables A Ay P r r r r A Ay P µ + + U. 5 r r r r e consider te system of generalized coordinates Q ( xy) T ( µ ) T () =. Tey depend upon te eigt parameters p = A A A A P P. Te corresponding differential equations are given by te system of Equation () wit Jacobi integral given by (). e consider te solutions of Equations () for wic C is zero. If we consider te solutions of Equation () given by (4) for some fixed parameters value p ten tere may exist anoter solution given by (5) wit anoter parameter value p ( µ A A A A P P ) T e ave and ( µ ) ( µ ) ( µ ) ( µ ) x = x A A A A P P y = y A A A A P P x = x A A A A P P y = y A A A A P P = close to p. (4) 4

5 Solution (5) will reduce to Solution (4) as ow we give te displacements p = p p and ( µ ) ( µ ) ( µ ) ( µ ) x = x A A A A P P y = y A A A A P P x = x A A A A P P y = y A A A A P P p p. ξ = Q Q (were Q ( x y ) T = and ξ ( ξ ξ ) T = ) (5) and i.e. µ = µ µ A = A A = A = A A x x A = A A P = P P A = A A P = P P (6) ξ = ξ = y y (6a) e consider p and ξ as small quantities of te same order. Ten we ave te following variational equations: ξ = ξ + ξ + n ξ + µ + A + A + A + A + P+ P xx xy xµ xa xa xa xa xp xp ξ = ξ + ξ n ξ + µ + A + A + A + A + P+ P (7) xy yy yµ ya ya ya ya yp yp wit te integral constructed from Equation () retaining te first order terms only we get C = x ξ + y ξ ξ ξ µ A A A A P P (7a) x y µ A A A A P P. t = Q t = x + y. e assume tat (5) Te modulus of momentary velocity on te orbit is defined as ( ) ( ) is not corresponding to te equilibrium state i.e. ( t) 0 and we furter assume tat ( ) 0 t on te wole orbit. Terefore x and y become te mobile co-ordinates. e will now use te mobile coordinate system to draw te periodic orbits by resolving one of te axis along te velocity vector X= ( xy ) T and te oter axis along te normal vector Y= ( yx ) T. In te new coordinate system we consider te transition matrix S as follows: Consider te first column of S as Y( t) r( t) = = te unit vector wic is normal to te orbit i.e. it is ortogonal to te vector s(t). t ( ) X ( t) s( t) ( t) So we ave S { rs } = = te unit vector wic is tangent to te orbit and is te last column of te matrix S. It can be easily verified tat = dim(r) = dim(s) = so tat dim(s) =. T T s s r s S s e ( ) T = = 0 and = = 0 It may be noted tat y x y x T y x r = s = S = and S = S =. ( t) x ( t) y ( t) x y ( t) x y ow we may furter define r = r = y x te first line of S t T T ( ) ( ) s = s = x y te last line of S. t ( ) ( ) 5

6 e write α te vector of local coordinates in te new coordinate system as follows: α = dim( ) = and dim( M ) = M were is displacement along te normal to te orbit and M is displacement along te tangent to te orbit. Ten te new coordinates are given by ξ = Sα = ( r s) = r + sm M α S ξ r ξ M s ξ Substituting tese values into te integral (7a) we ave = = = (8) ξ = S α + S α = r + r + sm + sm (9) C = ( M M ) + ( xy yx + xy xy ) A A A A ' A A A P P 0. ' A Equation (0) can be solved for Ṁ as P P µ µ M M = ( xy yx + xy xy ) ( µ µ A A A A A A A A P P P P ) Equations of motion () for te new coordinates are - S α = ξ S α S α or α = S F + F + F p p+ Ms were F p ( Fµ F FA FA FA FA FP FP ) and = xα yα Since F l αl α l xα + ny x x l + = l = to 8 α l y α + nx + y y l α = µα = α = A α = A α = A α = A α = P α = P = 0 and = 0 and F and F are te same as in our papers []-[]. α = terefore te equations of motion in normal and tangent co-ordinates can be written as M Ṅ can also be written as ( P ) r F F F P. (0) () = + + () ( P ). () M = M + s F + F + F P 6

7 = xxy xy xy + yyx + xy + yx n + xy xy n( xy xy ) ( xy xy)( n xy xy) x y ( )( ) µ A i + x y µ + yµ x + ( n + xy xy ) µ + xa y + i ya x + ( n + xy xy ) A i i i= A P + xa y + ya x ( n xy xy) A ( ) i + + i + xp y + yp x + n + xy xy P i= P + xp y + yp x + ( n + xy xy ) P + n + xy xy ( ). Tus we ave derived te equation in Ṅ wic possesses te remarkable property tat te normal coordinate () is independent of te tangent coordinate (M). Moreover instead of te differential equation of te second order () we can use te first order differential Equation (). If te investigated motion (4) is periodic ten te matrix S(t) can be taken as periodic and Equations () and () will ave te periodic coefficients at p Periodic Orbits For determining te periodic orbits te required equations of motion and te variational equations are given as: x = ny + (4i) x y y = nx + (4ii) 0 I J Z j = Z j Z j ( 0) = ej ( j J ) rf rf = (4iii) µ j = µ j gz j µ j ( 0) = 0 (4iv) υ 0 I J 0 P ( 0) 0 ( ) k Pk P k K k rf r F rf υ υ = + = = (4v) Pk µ P = µ ( ) ( 0) 0 k P k g υ Pk g P µ k Pk υ + =. (4vi) were Z( t ) is te matrix of solutions of a omogeneous system wit initial condition Z ( ) I υ P ( t) = a particular solution of te equations wit zero initial conditions i.e. ( 0) 0 k P k µ ( t) and ( t) Pk 0 J = and υ =. Te row-vector µ are te solutions of Caucy problem (iv) wit (vi) of (4). Te order of te above system is tirty-four. So for finding te new periodic motion it is necessary to integrate te system (4) of te differential equations from t = 0 and t = T. In te formulae (i) to (vi) of (4) it may be noted tat I J = (e e J ) Z = (Z Z J ) μ 0 ẏ 0 are known. = (μ μ J ) and te initial conditions x(0) y(0) ẋ ( ) and ( ) After solving te above equations of motion (i) and (ii) te variational Equations (iii)-(vi) of (4) and applying te predictor-corrector metod we ave determined te periodic orbits. e ave drawn te periodic orbits for te following: ) for fixed μ = 0.00 A = 0.0 A = 0.0 A = 0.00 A = 0.0 P = 0.0 and P = 0.0 (Figure ); ) for fixed μ = 0.00 A = 0.00 A = 0.0 A = 0.00 A = 0.0 P = and P = 0.00 (Figure ); ) for fixed μ = 0.00 A = 0.00 A = 0.00 A = 0.0 A = 0.0 P = 0.0 and P = (Figure 4); 4) for fixed μ = 0.00 A = 0.00 A = 0.00 A = 0.00 A = 0.00 P = and P = (Figure 5); 5) for fixed μ = 0.00 A = 0.00 A = 0.00 A = A = P = 0.00 and P = 0.00 (Figure 6). 7

8 Figure. Periodic orbits wen μ = 0.00 A = 0.0 A = 0.0 A = 0.00 A = 0.0 P = 0.0 and P = 0.0. Figure. Periodic orbits wen μ = 0.00 A = 0.00 A = 0.0 A = 0.00 A = 0.0 P = and P = 0.0. Figure 4. Periodic orbits wen μ = 0.00 A = 0.00 A = 0.00 A = 0.0 A = 0.0 P = 0.00 and P =

9 Figure 5. Periodic orbits wen μ = 0.0 A = 0.00 A = 0.00 A = 0.00 A = 0.00 P = and P = Figure 6. Periodic orbits wen μ = 0.00 A = 0.00 A = 0.00 A = A = P = 0.00 and P = In eac figure we ave drawn 5 periodic orbits corresponding to different values of. Tese orbits ave been numbered 4 and 5 corresponding to different values of. Te above analysis is summed up in Table. By taking bot te primaries as radiating triaxial rigid bodies te difference in te beavior of te values of is obvious. 5. Conclusions Karimov and Sokolsky [] ave studied periodic orbits in te restricted tree body problem by giving te displacements along te normal and te tangent to te orbit at te mobile co-ordinates. Tey ave taken bot te primaries as point masses wile in tis paper besides taking bot te primaries as triaxial rigid bodies we ave also taken bot te primaries as source of radiation pressure as well. In tis paper we ave again determined 9

10 Table. Te summary analysis of te periodic orbits. μ = 0.00 Oblate body Figure Triaxial body Figure Triaxial body Figure 4 Triaxial body Figure 5 Triaxial body Figure 6 alues of Energy Constant A = 0.0 A = 0.00 A = 0.0 A = 0.0 P = 0.0 P = 0.0 A = 0.00 A = 0.00 A = 0.0 A = 0.0 P = P = 0.0 A = 0.00 A = 0.0 A = 0.00 A = 0.0 P = 0.0 P = A = 0.00 A = 0.00 A = 0.00 A = 0.00 P = P = A = 0.00 A = 0.00 A = A = P = 0.00 P = five periodic orbits in a family for fixed value of te mass parameter μ te triaxial parameters A A A A and te radiation parameters P and P wit varying energy constant. e ave observed te following effects on te periodic orbits and on te energy constant due to triaxial rigid bodies and radiation pressure if we compare it wit te results of Karimov and Sokolsky [] and our papers []-[]. ) Te energy constant increases in a family (for Figure and Figure ) ten it decreases (for Figures 4-6) for fixed trixial parameters A A A A and radiation parameters P and P. ) As we increase te radiation parameters P and P te energy constant increases wereas te periodic orbits srink a little. ) Te periodic orbits go away from te libration point L 4 as we increase triaxial parameters A A A A and radiation parameters P and P wereas energy constant decrease. e ave investigated te family up to te member wic touces te point L 4. It is observed tat te families of periodic orbits in Karimov and Sokolsky [] terminate at bot te triangular equilibrium points simultaneously wile in our case tese families are non-symmetrical so tey may continue. Acknowledgements e are tankful to te Centre for Fundamental Researc in Space Dynamics and Celestial Mecanics (CFRSC) Deli and te Deansip of Scientific Researc College of Science in Zulfi Majmaa University KSA for providing all te researc facilities in te completion of tis researc work. References [] Mittal A. Aggarwal R. and Batnagar K.B. (0) Periodic Orbits around L 4 in te Potogravitational Restricted Problem wit Oblate Primaries. SEAS 6t International Conference Proceedings on Optics Astropysics and Astrology Article ID: [] Mittal A. Iqbal A. and Batnagar K.B. (008) Periodic Orbits Generated by Lagrangian Solutions of te Restricted Tree-Body Problem en One of te Primaries Is an Oblate Body. Astropysics and Space Science ttp://dx.doi.org/0.007/s [] Mittal A. Iqbal A. and Batnagar K.B. (009) Periodic Orbits in te Potogravitational Restricted Problem wit te Smaller Primary an Oblate Body. Astropysics and Space Science ttp://dx.doi.org/0.007/s [4] Carlier C.L. (899) Die Mecanik des Himmels. alter de Gryter and Co. Berlin and Leipzig. [5] Plummer H.C. (90) On Periodic Orbits in te eigborood of Centres of Liberation. Montly otices of te Royal Astronomical Society ttp://dx.doi.org/0.09/mnras/6..6 [6] Riabov U.A. (95) Preliminary Orbits Trojan Asteroids. Soviet Astronomy

11 [7] Szebeely. (967) Teory of Orbits: Te Restricted Problem of Tree Bodies. Academic Press ew York. [8] Deprit A. and Henrard J. (968) Advances in Astronomy and Astropysics. Academic Press ew York London. [9] Markeev A.P. and Sokolsky A.G. (975) Investigation of Periodic Motions near te Lagrangian Solutions of Restricted Tree-Body Problem. Publ. Inst. of Appl. Mat. Acad. Sci. Moscow. [0] Hadjidemetriou J.D. (984) Periodic Orbits. Celestial Mecanics ttp://dx.doi.org/0.007/bf0586 [] Karimov S.R. and Sokolsky A.G. (989) Periodic Motions Generated by Lagrangian Solutions of te Circular Restricted Tree-Body Problem. Celestial Mecanics and Dynamical Astronomy ttp://dx.doi.org/0.007/bf [] Taqvi Z.A.A.R. and Iqbal A. (006) on-linear Stability of L 4 in te Restricted Tree-Body Problem for Radiated Axes Symmetric Primaries wit Resonances. Bulletin of Astronomical Society of India 5-9. [] Abouelmagd E.I. Alotuali M.S. Guirao J.L.G. and Malaika H.M. (05) Periodic and Secular Solutions in te Restricted Tree-Body Problem under te Effect of Zonal Harmonic Parameters. Applied Matematics & Information Sciences [4] Perdios E.A. Kalantonis.S. Perdiou A.E. and ikaki A.A. (05) Equilibrium Points and Related Periodic Motions in te Restricted Tree-Body Problem wit Angular elocity and Radiation Effects. Advances in Astronomy [5] Jain M. and Aggarwal R. (05) A Study of on-collinear Libration Points in Restricted Tree-Body Problem wit Stokes Drag Effect en Smaller Primary Is an Oblate Speroid. Astropysics and Space Science 58-8.