 International Journal of Astronomy and Astrophysics, 2013, 3, 29-33 http://dx.doi.org/10.4236/ijaa.2013.31004 Published Online March 2013 (http://www.scirp.org/journal/ijaa) Approximate Kepler’s Elliptic Orbits with the Relativistic Effects Leilei Jia Department of Electrical Engineering, Guilin College of Aerospace Technology, Guilin, China Email: jiall@guat.edu.cn Received December 10, 2012; revised January 13, 2013; accepted January 21, 2013 ABSTRACT Beginning with a Lagrangian, we derived an approximate relativistic orbit equation which describes relativistic correc- tions to Keplerian orbits. The critical angular moment to guarantee the existence of periodic orbits is determined. An approximate relativistic Kepler’s elliptic orbit is illustrated by numerical simulation via a second-order perturbation method of averaging. Keywords: Kepler’s Elliptic Orbits; The Relativistic Kepler Problem; Unboundedness; Averaging 1. Introduction Kepler problem is one of the fundamental problems of orbital mechanics [1,2], which has been studied widely [3-5]. It is regarded as a special case of two-body prob- lems , where one body is assumed to be fixed at the origin-say, for example, it is so massive, like the Sun, that to the first approximation it does not move. The Ke- pler’s elliptic orbit is a conic section of the Kepler’s equation in polar coordinates with the form ,0 11coscrrefe, (1) where is the eccentricity and the angle ef is often called the true anomaly. Such elliptic orbits are of im- portance on describing dynamics of orbital mechanics in celestial mechanics and astrophysics. When dealing with particles moving at speed close to that of light it may be important to take into account the relativistic effects [7-12]. There have been several at- tempts to obtain the orbit solution for a classical relativ- istic two-body system interacting electromagnetically, and the concentric circular motion of two classical rela- tivistic point charges interacting electromagnetically had been described [13-16]. In this paper, using a perturba- tion techniques of averaging we will give the approxi- mate Kepler’s ellipse orbits for the Kepler problem with the special relativistic effects. In our results, we will show that once the relativistic contribution to Kepler problem is considered, the Kepler’s ellipse orbit may be destroyed. However, they perhaps maintain the original characteristics for a long time. The paper is organized as follows. Firstly, the Lagran- gian equations of motion of the relativistic Kepler prob- lem are deduced, and the elliptic periodic orbits and un- bounded orbits of equations are determined. Secondly, by the near-identity transformation, a good approximation of the Kepler’s elliptic orbits is obtained via averaging of the angle. An example is given to illustrate the applica- tion of the result. Finally, we conclude our results. 2. Periodic and Unbounded Orbits of the Relativisti c K e pl e r P roblem Under relativistic effects, a particle of mass orbiting a central mass mM is commonly described by the Lagrangian in the polar coordinates [17-21] 2222GMmLmccrr r  (2) where is Newton’s universal gravitational constant and is the speed of light in vacuum. Then the La- grangian equations of motion are given by Gc222 22222 2dddd1,1LL rtr rtrr crrr c  (3) 22222dd 0.dd1LL rttrr c (4) At this moment it is convenient to introduce the relativistic linear momentum  pCopyright © 2013 SciRes. IJAA L. L. JIA 30 22221rprr c (5) and Equation (4) implies the conservation of the rela- tivistic angle momentum , an arbitrary constant of integration, 22222.1rrrc (6) By a simply algebraic computation, we have 22 22 222 22 2,.crprcr prcrcr pr (7) Substituting (7) into (3) and together with (5), it yields that 22 22 222222 22 2,.crprcr prcGprrcrprM (8) Note that Equation (8) have periodic orbits if and on- ly if the relativistic angle momentum  is large enou- gh, precisely, :cGM c . In fact, the derivation of the relativistic linear momentum always be nega- tive when the opposite direction of inequality holds, since p222 2222111.cprcr prGMrGM For example, the mass of Sun, the Newton’s universal gravitational constant and the speed of the light are taken to be and , respectively, then the periodic orbits exist only for 3011 3121.989 10kg,6.670 10mkgsMG 812.99810 ms11 14.425 10 s.cSince the change of the polar coordinates preserve the symplectic form, Equation (8) retains the Hamilton structure with the Hamiltonian 22 22 2,.cHrpcr prrGMr (9) The curve of level set with the Hamiltonian 22222,, 1,HrphhcGMc  defines the “energy” of the Kepler system (8). When 22221hc GMc2 , the curve of level set reduces to an elliptic equilibrium point 2222,1GMrp cc ,0 of Equations (8), which is corresponding to the circle of Keplerian orbits of the form (1) with . At the same time, every curve of level set with 0e2 222221,hcGMc c is corresponding to a periodic orbit of Equation (8). In case of 2,hcr, the orbits become unbounded and insect the -axis only one time. The orbits in the phase plane ,rp for Equation (8) are depicted in Figure 1 using the parameters mentioned above. 3. Approximate Kepler’s Elliptic Orbits In previous section, we find that the large relativistic an- gle momentum  is necessary and sufficient to guaran- tee the existence of the periodic orbits. At the same time,  also as a constant of integration can be taken arbitrar- ily large. Consequently, in this section we will assume that  is so large that 2222 1.GMc In the following, with this assumption by the method of averaging, we will show that for a long time the orbit on the ,r plane is an approximate Kepler elliptic orbits. The averaged method has been used widely [23-26]. Together with (7) and (8), by successive applications of the chain rule, we get 2dddddddrprrptt rd , (10) 2222d1dd 1dddprGMpptt rcc. (11) So it follows that 2222dd1,ddccc crrr rrrr r  (12) where 2crGM. Let 1, ,crr , then we obtain that 2222d11d1.  (13) Copyright © 2013 SciRes. IJAA L. L. JIA Copyright © 2013 SciRes. IJAA 31 010002000300040005000510805108rmpkg.ms1  Figure 1. Periodic and unboun de d orbits in the plane for the relativistic Kepler proble m. Let sin, cos.JJ  (14) The perturbed Equation (13) becomes    2222 22222 22cos1cossin 111sin1coscos1sin2 sin2.1sincos1sin2 sin2JJJJJJJOJJJJ       (15) the system (15) transforms into By the near-identity transformation 12,,,,UJJJUJ     210,2JO O2,   (17)  (16) where   1222,, 1cos 111sin1sinsin 222,,UJ JJJ JUJ 4    . 0,0 π2Je Similarly, for the equation, averaged to second-order, we obtain is easily solved by 32,π11 .228JeOO3   (19)  32110, 328JO O.   (18) The Equation (18) with the initial value L. L. JIA 32 Combining with the transformation (14) and (16), we have  1222222221sin1, ,sin, ,1cos11cos1228281cossincos 24.28 2284crJJUJ UJreeeOe   2          As an example, we illustrate our results for Mercury of our solar system which is described by the near-circular orbit. Mercury has the eccentricity by the classical Newton mechanics. The other parameters are taken as follows: 0.2056eNewton’s universal gravitational constant G = 6.670 × 10−11 m3·kg−1·s−2; the mass of the Sun M = 1.989 × 1030 kg; the speed of light G = 2.998 × 108 m·s−1; the relativis- tic angular moment μ = 10μc = 10 GM/c. An approximate Kepler elliptic orbit due to special relativity is illustrated in Figure 2. 4. Conclusion The relativistic angle momentum  determines the existence of periodic orbits. When  is smaller than the critical angle momentum c, the Kepler system (8) has no periodic orbits. For c, if the energy defined by (9) lies in a proper interval 2 222221,hcGMc c , then every orbit is closed and periodic; otherwise, it leads to the unbounded orbits. The approximate relativistic Kepler elliptic orbit is illustrated by numerical simulation via a second-order perturbation method of averaging, and it is valid only for timescale of the order of 21. 5. Acknowledgements This work is supported by the National Natural Science Einst einKepl er15000010000050000050 000100 00015000015000010000050000050000100000150000rcos mrsinm Figure 2. Relativistic orbit in a Keplerian limit (blue solid line), as described by Equation (13), compared to a corresponding Keplerian orbit (red dashed line) with 0. The approximate Kepler elliptic orbit due to special relativity is illustrated here for .0 100 Copyright © 2013 SciRes. IJAA L. L. 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