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,

= 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 + ecos( f - W ) }

(3.2)

Where in [2], Eq.(5.6)

m =

(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,

wr²

w₀ r₀²

(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 -

s²

c²u²

w²s²

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,

w₀ s₀²

æ
è

1 -

w₀² 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 =

2pa^3/2

(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.

××

= - c²u

+ 2

+ 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 -

2gm_G

sc²

ö
ø

1/2

(3.11)

So that in (3.10)

- c²u

= -

gm_G

s²

(3.12)

as derived in [2], Eq. (4.4) and where

g = Newton's constant of proportionality.
m_G = The mass of the gravitational source.

Inserting (3.12) into (3.10) gives,

××

= -

gm_G

+ 2

+ u²w²s

(3.13)

In (3.13) over one complete orbit, the mean values of

and

××

will be zero, s will become s_OR, w will become w_OR, and u becomes u_OR where,

s_OR = The mean value of s over one complete orbit.
w_OR = The mean value of w over one complete orbit.
u_OR = The mean value of u over one complete orbit.

so that (3.13) then becomes,

u_OR² w_OR² s_OR =

gm_G

s_OR²

(3.14)

u_OR is determined simply thus. The temporal rate at s is given by [2], Eq. (4.6), viz,

u =

æ
è

1 -

ö
ø

1/2

(3.15)

so that

u_OR =

æ
è

1 -

s_OR

ö
ø

1/2

(3.16)

w_OR is determined as follows. In one complete orbit the angle rotated through is

f_OR = 2p - W_OR

(3.17)

Where

W_OR = The single orbit perihelion precession angle.

From (3.5) this is given by

W_OR =

6pam₀

1 + e

(3.18)

and therefore the mean value of the angular rate over one orbit is,

w_OR =

f_OR

t_OR

æ
è

1 -

3am₀

1+e

ö
ø

(3.19)

Where

t_OR = 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

gm_G

s_OR²

4p²

t_OR²

æ
è

1 -

s_OR

ö
ø

s_OR

æ
è

3au₀

1+e

ö
ø

(3.20)

Re-arrangement into Kepler's format then yields,

s_OR² ( s_OR - 2a )

t_OR²

gm_G ( 1 + e )²

4p²( 1 + e - 3au₀ )²

(3.21)

Eq.(3.21) shows that, irrespective of the actual value of s_OR, 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, s_OR. This is shown as follows. In

erting 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

W_OR

and so (3.22) becomes

s |_{_{f = p}} =

p( 1 - e² )

1 - ecos( W²_OR )

(3.24)

Therefore, from (3.23) and (3.24)

s_OR

|^p
|_{_{f = 0}} =

p( 1 - e )

é
ë

1 +

{ 1 - cos( W²_OR ) }

ù
û

{ 1 - ecos( W²_OR ) }

(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 W_OR become zero, which results in (3.25) becoming

s_OR = p

(3.26)

so that (3.21) then becomes

p³

t_OR²

gm_G

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

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

3.1 Kepler's First Law.

3.2 Kepler's Second Law.

3.3 Kepler's Third Law.