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
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
c
r
r
ef



e 
, (1)
where is the eccentricity and the angle
e
f
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
m
M
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
G
c


222 2
2
222 2
dd
dd
1
,
1
LL r
tr rtrr c
r
rr c





 







(3)

2
2222
dd 0.
dd
1
LL r
tt
rr c











(4)
At this moment it is convenient to introduce the
relativistic linear momentum [22]
p
C
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L. L. JIA 30

2222
1
r
p
rr c 




(5)
and Equation (4) implies the conservation of the rela-
tivistic angle momentum

, an arbitrary constant of
integration,

2
2222
.
1
r
rrc

 



(6)
By a simply algebraic computation, we have
22 22 2
22 22 2
,
.
crp
r
cr pr
c
rcr pr













(7)
Substituting (7) into (3) and together with (5), it yields
that
22 22 2
2
2
222 22 2
,
.
crp
r
cr pr
cG
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
p


222 222
2
1
1
1.
c
p
rcr pr
GM
r











G
M







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

 
81
2.99810 ms 
11 1
4.425 10 s. 


c
Since the change of the polar coordinates preserve the
symplectic form, Equation (8) retains the Hamilton
structure with the Hamiltonian

22 22 2
,.
c
Hrpcr pr
r
 
GM
r
(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

22
22
,1
GM
rp 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 22222
1,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
22
22 1.
GM
c
 
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
2
d dd
dd dd
rpr rp
tt r
 

d

 , (10)
2
2
22
d1 dd 1
dd
d
prGMp
p
tt r
cc







. (11)
So it follows that
22
2
2
dd
1,
d d
ccc c
rrr r
rrr r



 


(12)
where 2
c
rG  M.
Let

1, ,
c
rr 

 ,
then we obtain that

22
2
2
d11
d


1.


 


(13)

L. L. JIA 31
0
100
0
200
0
300
0
400
0
500
0
 5  10
8
0
5  10
8
r
m
pkg.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


 



 
 

22
22 22
2
22 22
cos
1cossin 11
1sin
1coscos1sin2 sin
2.
1sincos1sin2 sin
2
JJJ
J
JJJ
O
JJJ
J

 

 


 


 
 

 

 

 




 










(15)
the system (15) transforms into By the near-identity transformation




1
2
,,
,,
UJ
JJ
UJ








 


 
  
 
2
1
0,
2
JO O
2
,

   

 (17)


(16)
where




  








12
2
2
,, 1cos 1
11
sin1sinsin 2
22
,,
UJ J
JJ J
UJ
 
4



 


 
 

 


 .


0,0 π2Je  
Similarly, for the equation, averaged to second-order,
we obtain is easily solved by



3
2
,
π11 .
228
JeO
O

3

 


 


(19)
 
32
11
0, 3
28
JO O.

   
  (18)
The Equation (18) with the initial value

L. L. JIA 32
Combining with the transformation (14) and (16), we have








 
12
22
2
2
222
1sin1, ,sin, ,
1cos11cos1
22828
1
cossincos 24.
28 2284
c
rJJUJ UJ
r
eee
O
e
 
 


 

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 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 1

.
5. Acknowledgements
This work is supported by the National Natural Science
Einst ein
Kepl e
r
15000
0
10000
0
5000
0
0
50 00
0
100 00
0
15
0
00
0
150000
10
0
00
0
 5
0
00
0
0
5
0
00
0
100000
15
0
00
0
rcos m
rsin


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



L. L. JIA 33
Foundation of China (Grant No. 11226130 and No.
11261013).
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