3  Kepler's Laws of Planetary Motion - Relativistic Domain Theory Applicability.

3.1  Kepler's First Law.

In [1], Kepler's first law is stated as follows,
"The planets describe ellipses with the Sun situated at a focus"

In [1], after analysis, this is slightly extended, to show that the planets describe conic sections about the Sun at one focus. It is well known that all gravitationally generated central orbits, in non-relativistic form, open or closed, describe conic sections given by,

L
r
 = 1 + ecos( f - f0 )
(3.1)
Where
L = The semi-latus rectum.
e  = Eccentricity.
r   = The radius vector to the focal point.
f  = The radial angle to the major axis.
f0 = A constant, (initial condition).

In the Relativistic Domain D1, it is shown in [2] Section 5, that a central orbit is also a conic section described thus, [2], Eq.(5.11),

m = 1
L
 { 1 + ecos( f - W ) }
(3.2)
Where in [2], Eq.(5.6)
m = 1
s
(3.3)
and in [2], Eq.(4.18)
s = r + a
(3.4)
and where in [2], Eq.(5.34),
W @ 3am0
1+e
 f + am0 sinf
1+e
(3.5)
In (3.2), (3.3), (3.4) and (3.5) the terms are defined thus,
W Orbit precession angle.
a Gravitational radius of the source.
m0    = Initial condition of m, (Inverse distance to orbit focus at perihelion).
The derivation of (3.2), (3.3), (3.4) and (3.5) is shown in [2] and in [3] it is shown that (3.4) is a relativistic effect within the gravitational source.

Because W is a variable, a precise re-statement of Kepler's first law, specifically for relativistic effects would therefore need to be as follows,

Äll gravitationally generated central orbits describe conic sections with the gravitational source situated at one focus, and in which precession of the orbit is a function of both the radial angle to the major axis and the eccentricity of the basic curve."

3.2  Kepler's Second Law.

In [1], Kepler's second law is stated as follows,
"The radius vector joining the Sun with a planet describes equal areas in equal times, i.e. the rate of description of sectorial area is constant".
Mathematically this is stated as,
dz
dt
 = h
2
 = wr 2
2
 = w0 r02
2
(3.6)
Where,
z = Swept area.
h = Swept area constant.
w = Angular rate
r = Radial distance.
w0 and r0 are initial conditions.
In [2] it is shown, [2], Eq.(5.4), that the relativistic version of the parameter h is given by,

h = 
ws2
æ
è 		1 - 
×
s 2
c2u2
  -  w2s 2
c2
 ö
ø 		1/2

 
  = 
w02 s02
æ
è 		1 -  w02 s02
c2
 ö
ø 		1/2

 
(3.7)
and thus the relativistic version of (3.6) becomes,
dz
dt
  =  h
2
  = 
w0 s02
2 æ
è 		1 -  w0 2 s02
c2
 ö
ø 		1/2

 
(3.8)
Where,
t = Time in D1.
and other terms are as defined above.

Kepler's second law therefore holds in Relativistic Domain Theory with the relationship modification as shown in (3.8).

3.3  Kepler's Third Law.

In [1], Kepler's third law is stated as follows,
"The cubes of the mean distances of the planets from the Sun are proportional to the squares of their times of revolution, i.e. if 2a is the major axis of the elliptic orbit and t is the periodic time, then t2 µ a3".
In [1] the mathematical relationship for this law is derived and quoted as,
t = 
2pa3/2
Ö
 
m
 
(3.9)
Where,
2a = The major axis.
t   = The periodic time, ( º t in the definition).

m   = A constant related to the gravitational force exerted by the source at the focus, (it is not the same as the parameter m used in this series of papers).

In the Relativistic Domain D1, to determine whether this law holds, the equation of motion applicable to a central orbit is used as the starting point. This is given by [2]. Eq. (3.17), viz.

××
s
 
 = - c2u du
ds
 + 2
×
s
u
 du
ds
 + u2w2s
(3.10)
Where
u = The temporal rate at s.
w = The angular rate.
From [2], Eqs. (4.6) and (4.7)
u = æ
è 		1 - 2gmG
sc2
 ö
ø 		1/2

 
(3.11)
So that in (3.10)
- c2u du
ds
 = - gmG
s2
(3.12)
as derived in [2], Eq. (4.4) and where
g    = Newton's constant of proportionality.
mG = The mass of the gravitational source.
Inserting (3.12) into (3.10) gives,

××
s
 
= -
gmG
s
+ 2
×
s
u
du
ds
+ u2w2s
(3.13)
In (3.13) over one complete orbit, the mean values of  
×
s
 
  and 
××
s
 
  will be zero, s  will become sOR, w will become wOR, and u becomes uOR where,



sOR = The mean value of s over one complete orbit.
wOR = The mean value of w over one complete orbit.
uOR = The mean value of u over one complete orbit.
so that (3.13) then becomes,
uOR2 wOR2 sOR = gmG
sOR2
(3.14)
uOR is determined simply thus. The temporal rate at s is given by [2], Eq. (4.6), viz,
u = æ
è 		1 - 2a
s
 ö
ø 		1/2

 
(3.15)
so that
uOR = æ
è 		1 - 2a
sOR
ö
ø 		1/2

 
(3.16)
wOR is determined as follows. In one complete orbit the angle rotated through is
fOR = 2p - WOR
(3.17)
Where
WOR = The single orbit perihelion precession angle.
From (3.5) this is given by
WOR = 6pam0
1 + e
(3.18)
and therefore the mean value of the angular rate over one orbit is,
wOR =  fOR
tOR
=  2p
tOR
æ
è 		1 - 3am0
1+e
 ö
ø
(3.19)
Where
tOR = The periodic time of the orbit.
Here m0 is as defined in this series of papers.
Substitution of (3.16) and (3.19) into (3.14) then gives
gmG
sOR2
=  4p2
tOR2
æ
è 		1 - 2a
sOR
ö
ø 		sOR æ
è 		1- 3au0
1+e
 ö
ø 		2

 
(3.20)
Re-arrangement into Kepler's format then yields,
sOR2 ( sOR - 2a )
tOR2
= gmG ( 1 + e )2
4p2( 1 + e - 3au0 )2
(3.21)

Eq.(3.21) shows that, irrespective of the actual value of sOR, Kepler's third law does not quite hold under relativistic conditions. The reason being the presence of the -2a term on the LHS of (3.21) and the -3am0 term on the RHS. Compounding this, there is a difference in the mean value of the radius vector, sOR. This is shown as follows.

Inserting the classical value for L into (3.2) and re-arranging gives,
s = p( 1 -e2 )
1 + ecos( f - W )
(3.22)
Where,
2p = The major axis length of the orbit.
Now, when f = 0, W = 0 and (3.22) reduces to,
s |f = 0 = p( 1 - e )
(3.23)
and when f = p, from (3.5),  W =  3pam0
1+e
 = WOR
2

and so (3.22) becomes
s |f = p = p( 1 - e2 )
1 - ecos( W2OR )
(3.24)
Therefore, from (3.23) and (3.24)
sOR| p
| f = 0 =
 p( 1 - e )  é
ë 		1 +  e
2
 { 1 - cos( W2OR ) } ù
û
{ 1 - ecos( W2OR ) }
(3.25)

As both halves of the orbit are identical, (3.25) also applies over the complete orbit.

Eq.(3.25) shows that the difference between the mean values of the radius vectors in Kepler's original third law and the relativistic version is solely due to the orbit precession angle.

Approximation to the non-relativistic version, Kepler's original, is obtained by allowing c ® ¥ so that both a and WOR become zero, which results in (3.25) becoming

sOR = p
(3.26)
so that (3.21) then becomes
p3
tOR2
=  gmG
4p2
(3.27)
as in (3.9).

The combination of Eqs.(3.21) and (3.25), although not equivalent to Kepler's third law is very close to it and represents the relationship for an elliptic orbit. Putting e = 0 produces the relationship for a circular orbit.

In view of the smallness of the discrepancy of (3.21) and (3.25) from (3.9), a re-statement of this law for a relativistic planetary orbit is unnecessary provided the modified relationships are used as appropriate



R6 Version 1.0.0
Ó P.G.Bass, November 2008
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