Mehmet Pakdemirli* Precession of a Planet with the Multiple Scales Lindstedt Poincare Technique (2)

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1 Z. Natrforsch. 05; aop Mehmet Pakemirli* Precession of a Planet with the Mltiple Scales Linstet Poincare Techniqe DOI 0.55/zna Receive May, 05; accepte Jly 5, 05 Abstract: The recently evelope mltiple scales Linstet Poincare (MSLP) techniqe is sccessflly applie to a mathematical moel of planet motion. The eqation is originally evelope to precisely nerstan the orbital motion of the planet Mercry aron the Sn an the precession of the orbit e to the relativistic effects. The qaratic nonlinear eqation is solve by the classical Linstet Poincare metho (LP) an then by the newly evelope mltiple scales Linstet Poincare metho (MSLP). Both approximate soltions are contraste with the nmerical simlations. When the relativistic effects are small, all three soltions coincie with each other. When the pertrbation effects are increase, the MSLP soltions agree better with the nmerical soltions than the LP soltions. The precession of the perihelion of the planet is calclate an compare for the approximate soltions. Keywors: Linstet Poincare Metho; Mltiple Scales Linstet Poincare Metho; Pertrbation Analysis; Planet Motion. Introction Using the Einstein gravitation theory, the precession of the planet Mercry was moelle with the ifferential system [, ] + = + ε a () (0) = b, (0) = 0 () *Corresponing athor: Mehmet Pakemirli, Applie Mathematics an Comptation Center, Celal Bayar University, 4540, Mraiye, Manisa, mpak@cb.e.tr where is the angle coorinate an is the reciprocal of the imensionless isplacement. = (3) r / r r is the istance between Mercry an Sn an r is the average istance ( r = m). The remaining imensionless qantities are as follows: GMr 3GM a =, ε = h c r where G is the niversal gravitational constant, M is the mass of the Sn, h is the anglar momentm per nit mass for Mercry, c is the spee of light in the vacm, b is the reciprocal of the isplacement when the planet is closest to Sn. Initially, the planet is assme to be in the closest istance to Sn (perihelion). For the planet Mercry, a = 0.98, b =.0. Accoring to the theory, the qaratic nonlinear term moels the small pertrbations from the orbit of the Mercry an has the orer of magnite ε O(0 7 ) or more precisely ε = []. Other than the implicit integral form that is evalate nmerically or expressions in Jacobi elliptic fnctions [], the moel oes not possess a simple analytical close form soltion. The ifferential system is solve first by the Linstet Poincare techniqe (LP) that is evelope by the astronomers to nerstan the orbital motion of the planets an small pertrbations in the trajectories. Next, the recently evelope mltiple scales Linstet Poincare techniqe (MSLP) [3 ] is employe in search of approximate analytical soltions. The metho combines the elimination mechanisms of seclar terms of both the classical Linstet Poincare techniqe an the classical mltiple scales techniqe an increases the possibility of sccess for obtaining niform soltions for the nonlinear ynamical systems. Another featre of the metho is the incorporation of an nsal freqency expansion sggeste by H [7] an H an Xiong [8], which leas to the niformly vali soltions for the strongly nonlinear systems. In a pioneering work, for a classical anharmonic oscillator with rescale pertrbation series, soltions vali for strongly nonlinear systems were also obtaine [9] by employing a similar reasoning. Finally, the (4) Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

2 M. Pakemirli: Precession of a Planet system is also solve nmerically by sing a variable step size Rnge Ktta algorithm. As in the case of the planet Mercry, when the pertrbation parameter is qite small, all three soltions coincie with each other an the precession angles are the same. However, when the pertrbation parameter is increase, the MSLP soltions are in close agreement with the nmerical soltions, whereas the LP soltions iverge from the others. The precession angles also start ifferentiating from each other. Pertrbation Analysis In this section, the system () an () is solve by the Linstet Poincare metho first an then by the mltiple scales Linstet Poincare metho.. The Linstet Poincare Metho In this metho, first the coorinate transformation is inserte into the eqation = (5) + = a+ ε () where is an introce transformation freqency. The epenent variable an the freqency are expane as the following pertrbation series: ( ; ε) = ( ) + ε ( ) + ε ( ) + (7) 0 = + ε + ε + (8) By applying into () an separate with respect to similar orers, the following eqations are obtaine. 0 + = = 0 a, (0) b, (0) = 0 (9) = 0 + =, (0) 0, (0) = 0 (0) = +. () 0 Since at the last level of approximation, only the seclar terms will be eliminate, an will not be calclate openly, one oes not nee the conitions at this level [0]. At the first-orer, the soltion is = ( b a)cos + a 0 () which pon sbstittion into the next level an elimination of the seclar term yiels = a (3) The soltion at this level is = a + ( b a) ( cos ) + ( b a) ( cos ) 3 (4) Finally, at the last level, 0 an are sbstitte an only the seclarities are eliminate. 5 = a ( b a ) (5) Combining the terms an retrning back to the original variables yiels = ( b a)cos + a+ ε a + ( b a) ( cos ) 3 () + ( b a ) ( cos ) + O ( ε ) = ε ε + 5 a a ( b a ), (7) which is the esire LP soltion for the problem. Smith [] gave a pertrbation soltion of the problem sing both the Krylov Bogolibov techniqe an the mltiple scales techniqe which is = ( b a)cos + a+ O ( ε) (8) = εa (9) It is to be note that the above soltion is a first approximation of the fnction an the freqency of the LP techniqe.. The Mltiple Scales Linstet Poincare Metho In this metho, similar to the LP metho, initially, a coorinate transformation is introce. = (0) an sbstitte into the original eqation Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

3 M. Pakemirli: Precession of a Planet 3 ε + = a + () As in the classical mltiple scales metho, the ifferent coorinate scales T =, T = ε, T = ε () 0 are efine an the associate erivatives are written in terms of the new variables = D + D + D + = D + DD + ε ( D + D D ) + ε ε, ε (3) where D n = / T n. The epenent variable is expane in a series in terms of the new inepenent variables ( ; ε) = ( T, T, T ) + ε ( T, T, T ) + ε ( T, T, T ) 0 (4) Instea of the transformation freqency, the normalise freqency is expane in this metho [3 ]. To be consistent, the parameter a shol also be expane in a similar manner. = ε ε + (5) a= a ( ε ε ) + () Sbstitting all into (), separating with respect to orers yiels ( D + ) = a, (0) = b, D (0) = 0 (7) The initial conitions ictate b (0) = b a, β (0) = 0 (33) Sbstitting (30) into the next level an eliminating the seclarities reqire i D B+ B+ ab = 0 (34) The metho gives priority to the elimination mechanism of LP first. Hence, pon selecting D B = 0, if is real, the choice is amissible which is exactly the case in (34) an hence B = BT ( ), = a (35) The soltion at this level is = E( T, T )exp( it ) + cc ( B exp( it ) + cc) ( BB a ) where The real form of is (3) E = e exp( i γ) (37) = + γ b e cos( T ) b cos(t + β) a The initial conitions reqire (38) ( D + ) = D D + + a, (0) = 0, ( D + D )(0) = (8) a e (0) = ( b a ), γ(0) = 0 (39) 3 ( D + ) = D D ( D + D D ) a The soltion at the leaing orer is (9) = B( T, T )exp( it ) + cc + a (30) 0 0 where cc stans for the complex conjgates an B is given by the following polar form. B = b exp( i β) (3) At the final level, only seclarities are eliminate. Sbstitting (3) an (30) into (9) an eliminating seclarities reqires 0 i D E i DB+ E+ B+ BB 3 a + B + ae = 0 (40) Withot loss of generality, D E = 0 can be selecte. For D B = 0, trns ot to be real an LP mechanism of eliminating seclarities is sfficient. Hence The soltion in real form is = b cos( T + β) + a (3) = = = E const, B const, b a, β = 0, γ = 0 (4) Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

4 4 M. Pakemirli: Precession of a Planet The freqency expansion now reas ε 5 = εa + a ( b a ) (4) It is to be note that now appears on both sies an has to be solve. Combining all the information an solving for the real an positive vale, the final approximate soltion in terms of the original variables an parameters is ε = ( b a)cos + a+ a + ( b a) ( cos ) 3 + ( b a ) ( cos ) + O ( ε ) (43) 5 = εa+ ( εa) 4ε a + ( b a) (44) Now, the freqency is slightly ifferent than the one obtaine by the LP metho. Frthermore, while the coefficient of the correction term is ε in the LP, here, the coefficient is ε/. It will be shown in the next section by comparing with the nmerical simlations that these slight ifferences in the freqencies an coefficients lea to more precise soltions Figre : Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = ) Nmerical LP MSLP 3 Comparisons The orinary ifferential system () an () is solve nmerically by employing an aaptive step size Rnge Ktta algorithm an compare with the analytical soltions. In Figre, the analytical soltions of both methos are contraste with the nmerical ones for the planet Mercry for ε = All soltions coincie with each other. When the pertrbation parameter is of orer, oscillation type soltions cannot be retrieve nmerically. If one assmes that the relativistic effects on the motion became larger somehow, the pertrbation parameter shol be increase from its originally calclate vale. In Figre, the pertrbation parameter is again small, bt relatively larger than its original vale, that is ε = While the nmerical an MSLP soltions agree with each other, LP soltions can be istingishe from both of them. In Figre 3, the LP, MSLP, an the one term KB soltions (i.e. 8 an 9) are contraste with the nmerical Figre : Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.05). soltion for a fairly large vale of the pertrbation parameter. The MSLP an the nmerical soltions agree well with each other. The qantitative iscrepancy is observe for the LP soltion. For the one term KB soltion given in [], there is both qalitative an qantitative iscrepancy compare to the other soltions. Finally, the parameter is increase p to the limit of oscillatory soltions where ε = 0. in Figre 4. While all three soltions are separate from each other, the MSLP better preicts the maximm amplites an freqencies compare to the LP. It is note that the motion of the planet is a qasiperioic motion becase the orbit itself oes not remain Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

5 M. Pakemirli: Precession of a Planet LP MSLP KB Nmerical Table : Precession angles for varios pertrbation parameters. Precession angle/centry ε KB LP MSLP Figre 3: Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.) Nmerical LP MSLP Figre 4: Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.). fixe in the space rather the major axis of the ellipse rotates within time. The perihelion ths precesses by an increibly small amont for which the accmlate anglar vale can be measre in a time interval of a centry. For one perio of motion, the precession angle in raians is [] PA = π( ) rev (45) where the freqency is sbstitte from (9) for KB, from (7) for LP an from (44) for MSLP. For the planet Mercry, this is a rather small qantity an the accmlate angle for a centry is calclate as follows: PA ra rev 355 ays = π( ) centry rev 88 ays centry 80 eg 300 s π ra eg (4) where the planet Mercry completes one revoltion in approximately 88 earth ays. All three methos give the same precession angle of abot 40 s of angle for the pertrbation parameter of ε = which is a rather small qantity (Tab. ). This vale is compatible with the astronomical observations []. When the pertrbation parameter is increase, first, the KB soltion ifferentiates from the others, an then, all soltions ifferentiate slightly from each other for a sfficiently large parameter. Base on the figres, the highest precision is obtaine by the MSLP. 4 Conclsion A ifferential system erive sing Einstein gravitation theory is consiere. The moel incorporate the relativistic effects on the orbital motion of the planet Mercry aron the Sn. Approximate soltions are fon sing the classical LP techniqe commonly se in astronomy an the recently evelope mltiple scales LP techniqe. The approximate analytical soltions are contraste with the nmerical soltions. For the stanar pertrbation parameter of ε = for the planet Mercry, all three soltions agree well with each other. While the moel is originally evelope for Mercry, it can be applie to other planets also. It may happen that the pertrbation parameter, which represents the relativistic effects, may become mch larger than the reporte vale, an hence, the parameter is increase p to the limit of oscillatory soltions. It is fon that the MSLP metho preicts the isplacements better compare to the LP metho. In the previos sties [3 ], the MSLP is sccessflly applie to the strongly nonlinear systems procing compatible soltions with the nmerical simlations. In this sty, the strong nonlinear system oes not proce oscillatory soltions, an hence, the pertrbation parameter is increase near to the oscillatory limiting vale. Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

6 M. Pakemirli: Precession of a Planet References [] D. R. Smith, Singlar-Pertrbation Theory, Cambrige University Press, New York 985. [] H. Golstein, Classical Mechanics, Aison-Wesley, New York 980. [3] M. Pakemirli, M. M. F. Karahan, an H. Boyacı, Math. Compt. Appl. 4, 3 (009). [4] M. Pakemirli an M. M. F. Karahan, Math. Methos Appl. Sci. 33, 704 (00). [5] M. Pakemirli, M. M. F. Karahan, an H. Boyacı, Math. Compt. Appl., 879 (0). [] M. Pakemirli an G. Sarı, Math. Compt. Appl. 0, 37 (05). [7] H. H, J. Son Vib. 9, 409 (004). [8] H. H an Z. G. Xiong, J. Son Vib. 78, 437 (004). [9] K. Banerjee, J. K. Bhattacharjee, an H. S. Mani, Phys. Rev. A. 30, 8 (984). [0] A. H. Nayfeh, Introction to Pertrbation Techniqes, John Wiley an Sons, New York 985. Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM

Theorem (Change of Variables Theorem):

Theorem (Change of Variables Theorem): Avance Higher Notes (Unit ) Prereqisites: Integrating (a + b) n, sin (a + b) an cos (a + b); erivatives of tan, sec, cosec, cot, e an ln ; sm/ifference rles; areas ner an between crves. Maths Applications: