Restricted Three-Body Problem in Different Coordinate Systems

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1 Applied Mathematics Pblished Online September ( Restricted Three-Body Problem in Different Coordinate Systems II-In Sidereal Spherical Coordinates System Mohammed A. Sharaf * Aisha A. Alshaery Department of Astronomy Faclty of Science King Abdlaziz University Jeddah KSA Department of Mathematics Faclty of Science King Abdlaziz University Jeddah KSA * sharaf_adel@hotmail.com alshaary@hotmail.com Received Jly ; revised Agst ; accepted Agst 7 ABSTRACT In this paper of the series the eqations of motion for the spatial circlar restricted three-body problem in sidereal spherical coordinates system were established. Initial vale procedre that can be sed to compte both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedre was illstrated by nmerical example and graphical representations of the variations of the two sidereal coordinate systems. Keywords: Spatial Restricted Circlar Three Body Problem; Reglarization; Coordinate Transformations. Introdction In a previos commnication to this jornal [] hereafter referred to as Paper I we started or stdies towards establishing new differential eqations for the different forms of the three-body problem sing some important coordinate systems. By this we aims at obtaining differential eqations (see [] for details) which are: ) Reglar; ) Sitable for the geometry to which they referred; 3) Prodcing slow variations in the coordinates dring the orbital motion a property which prodces more stable nmerical integration procedres. In Paper I the eqations of motion for spatial restricted circlar three body problem in cylindrical coordinates system was established together with a comptational algorithm that can be sed to compte both the cylindrical and Cartesian coordinates and velocities. In the present paper the eqations of motion for the spatial circlar restricted threebody problem in sidereal spherical coordinates system were established. Initial vale procedre that can be sed to compte both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedre was illstrated by nmerical example and graphical representations of the variations of the two sidereal coordinate systems.. Circlar Restricted Three-Body Problem in Sidereal System If two of the bodies say m and m in the three-body * Corresponding athor. problem move in circlar coplanar orbits abot their common center of mass and the mass say m 3 of the third body is too small to affect the motion of the other bodies the problem of the motion of the third body is called the circlar restricted three body problem. The two revolving bodies are called the primaries; their masses are arbitrary bt have sch internal mass distribtions that they may be considered point masses. The eqations of motion of the third body in a dimensionless sidereal (inertial) coordinate x yz system with the mean motion n are [] V x () x V y () y V z (3) z V V x y z is given as V (4) r r denotes the mass of the smaller primary when the total mass of the primaries has been normalized to nity. r xx y y z ; i (5) i i i and ri ; i are the distances of the third body from the primaries which are located at xi yi; i these coordinates are fnctions of the time t and are Copyright SciRes.

2 95 M. A. SHARAF A. A. ALSHAERY given as x cos t; x cos t; y sin t; y sin t 3. The Eqations of Motion in Sidereal Spherical Coordinate System Corresponding to the Cartesian sidereal coordinate system x yz the coordinate system related to the system x yz by certain transformation is also called sidereal coordinate system. In this respect the system 3 of Eqation (7) is called sidereal spherical coordinate system. In what follows we shall establish the differential eqations for the spatial circlar restricted three bodyproblem in sidereal spherical coordinate system. 3.. Coordinate and Velocity Transformations (6) x sin cos 3; y sin sin 3; z cos (7) sin sin x cos3sin coscos y sin sin 3cossin cos sin (8.) (8.) z cos sin (8.3) π π π (9) Inverse Transformations From Eqation (7) we have x y z tan ; 3 tan ; x y z y x () Differentiating the first and the third of Eqation () and the third of Eqation (7) with respect to the time t we get: ( xx yyzz ) ; z z x y ; () and are given in terms of x yz and x yz 3.3. The Eqations of Motion The kinetic energy of a particle of nit mass in the spherical coordinate system is T 3 sin () By sing the transformation eqations (Eqations (7)) the gravitational potential V cold be expressed in term of 3. dd t T T V ; j 3 j j j Conseqently we dedce for the eqations of motion in sidereal spherical coordinate system the forms 3 sin V (3.) 3 sin V (3.) V csc cot (3.3) V j are given as V V x V y V z ; j 3 x y z j j j j (4) x j y j and z j ; j 3 can be compted from Eqation (7) while V x V y and V z can be compted from Eqations ()-(3) so we get V cosq Q sin Q Q Q cos Q sin QQ V sin QQ cos cos QQ QQ sin Q Q Q Q QQ V Q 7 QQ QQ 3 Q cos t sin 3 3 cos sin Q3 cos tcos3sin Q4 sin tsin sin 3 3 xy yx x y Q5 costsincos 3 from the previos eqations. Q6 sin tsin 3sin 3 Q t 3 Copyright SciRes.

3 M. A. SHARAF A. A. ALSHAERY 95 sin sin Q7 t 3. This initial vale procedre sing sidereal spherical coordinate system will be described throgh its basic points inpt otpt and comptational steps. 4. Comptational Development Inpt: ) x y z x y z at t t 4.. Initial Vale Procedre ) the final time t tf In what follows we shall establish a procedre that can V 3) F xyz ; be sed to compte t t tf (say) both: x ) The spherical sidereal coordinates and velocities V V Fxyz ; F3xyz ; 3 3 and y z ) The Cartesian sidereal coordinates and velocities Otpt: ) j; j; j 3 t t tf x yzxyz. ) x yzxyz ; t t tf So sch procedre is a doble seflness compta- Comptational steps tional algorithm for which a differential solver can be ) Using the given vales x sed for the spherical sidereal six order system to obtain y z x y z at 3 t t 3. While the Cartesian sidereal coor- and the inverse transformations to compte the dinates and velocities x yzxyz initial vales are obtained by ; 3456 j j. V V V the sbstittions in the direct transformation formlae ) Using the partial derivatives ; ; (fnc- (Eqations (7) and (8)) rather than solving the six order 3 system of Eqations () () and (3). By this way great tions of j; j 3 ) to constrct the analytical forms time can be saved. of eqations of motion as first order system. Table. The vales of sidereal spherical coordinates and velocities. t t t 3 t 4 t 5 t 6 t Copyright SciRes.

4 95 M. A. SHARAF A. A. ALSHAERY Table. The vales of sidereal Cartesian coordinates and velocities. t xt yt zt x t yt zt ) Using the initial conditions j; j 3456 from step to solve nmerically the above differential system of step for j; j 6 t t tf (note that ). 4) Using j; j ; j 3 from step 3 and the direct transformations of Eqations (7) and (8) to compte nmerically x yz and x yz t t tf. 5) End. 4.. Nmerical Example Consider the initial vales x ; y ; z ; x.76848; y ;.5586; t 8.; f t.; applying the above procedre we get the reslts as displayed in Tables and Graphical Representations Figre illstrates the time variations of the two sidereal coordinate systems x yz (left) and (right). 5. Conclsion 3 In this paper of the series the eqations of motion for the spatial circlar restricted three-body problem in sidereal spherical coordinates system were established. Initial vale procedre that can be sed to compte both the spherical and Cartesian sidereal coordinates and velocities was also developed. The application of the procedre Copyright SciRes.

5 M. A. SHARAF A. A. ALSHAERY 953 Figre. The time variations of the two sidereal coordinate systems (x y z) (left) and ( 3 ) (right). was illstrated by nmerical example and graphical representations of the variations of the two sidereal coordinate systems. I) Jornal of Applied Mathematics in Press. [] V. Szebehely Theory of Orbits Academic Press New York 967. REFERENCES [] M. A. Sharaf and A. A. Alshaery Restricted Three- Body Problem in Cylindrical Coordinates System (Paper Copyright SciRes.