Download Relativistic Third Kepler Law for Circular Orbits

Relativistic Third Kepler Law for Circular Orbits
Jerry Hynecek1
Isetex, Inc.
ABSTRACT
In this paper the third Kepler’s law is derived for the circular orbits using two different
relativistic metrics and compared with the third Kepler’s law derived from the Newtonian
physics. It is found that the mass of the Sun is slightly underestimated when using the traditional
calculations, which may affect the calculations of trajectories of space probes trying to reach
other planets of our Solar system.
Key words: Kepler’s third law, Relativistic Kepler’s third law, Schwarzschild metric.
INTRODUCTION
The Kepler’s third law is a very important law for astronomers, which is used to
determine the mass of the planets and stars based on the gravitational constant measured here on
Earth and on the time of the planet’s full orbit completion. Since the time of the orbit can be
measured with a high precision and the radius of the orbit is also reasonably well known, the
mass of the centrally gravitating body can be found very accurately. This law is easily derivable
from the Newton inertial and gravitational laws for a circular orbit by equating the centrifugal
force with the gravitational force as follows:
v 2 M
 2 ,
r
r
(1)
where M is the mass of the centrally gravitating body and  the gravitational constant. By
realizing that the average velocity is the length of the orbit circumference divided by the time of
the orbit Eq.1 can be rewritten in the familiar third Kepler’s law form:
t nt2 
4 2 r 3
.
M
(2)
With advancements in the theory of gravity from Newtonian to Einstein’s General Relativity
Theory (GRT) it is thus natural to ask now how is this law changed and is it accurate enough to
determine, for example, the mass of our Sun with enough precision so that no large trajectory
errors are generated when the space probes are sent to investigate other planets of the Solar
system.
1
[email protected] © 1,12,2010 1 DERIVATION OF THE RELATIVISTIC KEPLER’S THIRD LAW
While there are several derivations of this law already published in the literature [1] [2], the
derivations are not very clear and are sometimes using questionable assumptions. In this
paragraph the Kepler’s third law will be derived very simply in a general fashion for the two key
cases: the Schwarzschild metric space-time and the new metric space-time.
The general metric line element for a non-rotating centrally gravitating body is:


ds 2  g tt (cdt ) 2  g rr dr 2  g d 2  sin 2 d 2 .
(3)
In the new metric derived elsewhere [3] the metric coefficients are: g tt  exp( Rs /  ) , g rr g tt  1 ,
and g    2 g tt , while for the Schwarzschild metric they are: g tt  1  Rs / r , g rr g tt  1 , and
g  r 2 . Considering for simplicity motion only in the equatorial plane the Lagrangian
describing such motion of a small test body is then as follows:
 cdt 
 dr 
 d 
L  g tt 
  g rr    g  
 .
 d 
 d 
 d 
2
2
2
(4)
The two first integrals corresponding to the time coordinate and the angular coordinate are:
 dt
g tt 
 d
 d 
g 
  ,
 d 

k,

(5)
where k and  are the constants of integration to be determined later. Considering that the
Lagrangian itself is also the first integral the following relation must hold:

dr  g tt 2  g tt
 g rr
 2  
g
d  g

 
 2
  c 2 k 2  Lg tt .
(6)


In the next step it is necessary to find an expression for  . The process is simplified by realizing
that the first term in Eq.6 is zero. By differentiating Eq.6 with respect to  , considering that the
2
orbits must be stable and also considering that for the Lagrangian it holds: L  c 2 k 2 the result is:
 c 2 k 2 g tt
,
 
g tt / g 
2
(7)
where the dot represents the derivative with respect to radial coordinate. By substituting for 
into the first integral for the angular coordinate in Eq.5 and considering that the coordinate
orbital time, which is the observable quantity, is found when the angle is set to:   2 , the
following result is obtained:
2 t0 
2
c
1
g tt
g  g 
.

g tt
g tt
(8)
This is a general formula that can be used for any metric describing the space-time of a nonrotating centrally gravitating body. For the Schwarzschild metric the result is:
t sch 
2
c
1
g tt
2 r 3  1  3 Rs / 2 r 
,

Rs  1  Rs / r 
(9)
where Rs is the Schwarzschild radius defined as: Rs  2 M / c 2 . For the Newton case Eq.9 is
simplified by neglecting the term Rs / r in comparison to unity with the result identical to
formula in Eq.2:
t nt 
2
c
2r 3
,
Rs
2
c
1
g tt
t sch
1

t nt
g tt
1  3 Rs / 2 r
.
1  Rs / r
(10)
For the new metric the result is:
t jh 
2 3
,
Rs
t jh
t nt

1
g tt
3
r3
,
(11)
where the physical distance  is a function of the coordinate distance r and is calculated from
the following differential equation:
d  e Rs / 2  dr .
(12)
The orbital time differences from the Newton orbit time are plotted in Fig.1. It is interesting to
note that both formulas converge to the gravitational red shift formula for large distances, but for
the Schwarzschild metric there is a zero shift at the three halves of the Schwarzschild radius.
This once again confirms a problem for the Schwarzschild metric already discussed elsewhere [4].
Instead of calculating the differences in orbit times, it is also possible to calculate the
differences in the mass of the Sun when the orbit time is actually made equal to the measured
orbit time for Earth. This results in the following corrections to the mass of the Sun:
M sch  M nt  0.99  10 8 M nt ,
M jh  M nt  57.72 10 8 M nt .
(13)
The corrections are not large, but not insignificant when precise space probe trajectories are
needed. Of course, these results need to be recalculated relative to the Sun-Earth Barycenter and
other corrections included such as the corrections from the influence of other planets of the Solar
system etc. Nevertheless, it is interesting to note that the Newton calculated mass of the Sun
3 slightly underestimates its true value. The new metric on the other hand results in the slightly
larger mass that generates more gravitational pull on the space probes that would be normally
anticipated.
Orbit Time Difference from Newton [sec]
100
10
1
jh( s Rs)
sch( s Rs)
0.1
0.01
3
110
4
110
6
110
7
110
8
110
9
10
110
110
11
110
12
110
13
110
r ( s)
Sun-Planet Distance [m]
Fig.1. Graphs of time differences from the Newton orbit time calculated according to the
Kepler’s third law and from the relativistic Kepler’s third law using the Schwarzschild and the
new metrics for a planet orbit around the Sun.
CONCLUSIONS
In this article the correct relativistic formula for the Kepler’s third law was derived and
applied to our Solar system. It was found that the mass of the Sun is slightly underestimated by
using the traditional Kepler’s third law. The accurate knowledge of the Sun’s mass is important
to have for accurate calculations of trajectories of space probes sent to other planets within our
Solar system or beyond its boundaries.
REFERENCES
1. R. Y. Kezerashvili and J. F. V´azquez-Poritz, “Deviations from Keplerian Orbits for
Solar Sails”, arXiv:0907.3311v1 [gr-qc] 20 July 2009.
2. B.M. Baker and G. G. Byrd, “Relativistic Kepler’s Third Law”, The Astronomical
Journal, 305, pp. 623-633, June 15 1986.
3. J. Hynecek, “Remarks on the Equivalence of Inertial and Gravitational Masses and on the
Accuracy of Einstein’s Theory of Gravity”, Physics Essays volume 18, number 2, 2005.
4. J. Hynecek, “The Galileo effect and the general relativity theory”, Physics Essays,
volume 22, number 4, 2009.
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