Approximate Kepler’s Elliptic Orbits with the Relativistic Effects

doi:10.4236/ijaa.2013.31004

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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: [email protected]

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 [6], 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 1

1cos









, (1)

where is the eccentricity and the angle

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

is commonly described by the

Lagrangian in the polar coordinates [17-21]

2222

GMm

Lmccrr 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



222 2

LL r

tr rtrr c

rr c







 









(3)



2222

dd 0.

LL r

rr c





















(4)

At this moment it is convenient to introduce the

relativistic linear momentum [22]

L. L. JIA



2222

rr c







(5)

and Equation (4) implies the conservation of the rela-

tivistic angle momentum



, an arbitrary constant of

integration,



2222

rrc









(6)

By a simply algebraic computation, we have

22 22 2

crp

cr pr

rcr pr























(7)

Substituting (7) into (3) and together with (5), it yields

that

22 22 2

222 22 2

crp

cr pr

rcrpr



















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



222 222

rcr pr



















G













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 312

1.989 10kg,6.670 10mkgsMG



 

2.99810 ms



11 1

4.425 10 s.







Since the change of the polar coordinates preserve the

symplectic form, Equation (8) retains the Hamilton

structure with the Hamiltonian



22 22 2

Hrpcr pr





(9)

The curve of level set with the Hamiltonian





22222

,, 1,HrphhcGMc





 





defines the “energy” of the Kepler system (8). When

2222

1hc GMc2



 ,

the curve of level set reduces to an elliptic equilibrium

point



rp cc



 









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 22222

1,hcGMc c











is corresponding to a periodic orbit of Equation (8). In

case of



2,hc









, 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

22 1.





In the following, with this assumption by the method of

averaging, we will show that for a long time the orbit on

the







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

rpr

tt r









 , (10)

dd 1

prGMp

tt r















. (11)

So it follows that

ccc c

rrr r







 





(12)

where 2



M.

Let



1, ,





 ,

then we obtain that



d11





1.





 





(13)

L. L. JIA

100

200

300

400

500



pkg.ms



 

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





 



 



22 22

cos

1cossin 11

1sin

1coscos1sin2 sin

1sincos1sin2 sin

JJJ



 



 



 





 







 







 



























(15)

the system (15) transforms into By the near-identity transformation























 





 

  

 

JO O



 

 



 (17)



(16)

where









  

















,, 1cos 1

sin1sinsin 2

UJ J

JJ J

 







 





 

 



 









0,0 π2Je





Similarly, for the equation, averaged to second-order,

we obtain is easily solved by







π11 .

228

JeO





 





 





(19)

 

0, 3

JO O.



 

 



 (18)

The Equation (18) with the initial value

L. L. JIA

Combining with the transformation (14) and (16), we have















 

222

1sin1, ,sin, ,

1cos11cos1

22828

cossincos 24.

28 2284

rJJUJ UJ

eee

 

 





 





 

 





 

 



 



 

 



 

 

























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 m

3·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 22222

1,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 2



5. Acknowledgements

This work is supported by the National Natural Science

Einst ein

Kepl e

15000

10000

5000

50 00

100 00

150000



100000

rcos m

rsin



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





L. L. JIA 33

Foundation of China (Grant No. 11226130 and No.

11261013).

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