5  Planetary Orbits in D1 - Equation of the Orbit and its Solution.

The equation of a planetary orbit can be derived in either the axes of D1 or those of D0. In the former it is generally known as Einstein's Equation of Planetary Motion. The latter is derived in [2] from the metric of the space-time of the General Theory. The former is derived here from foregoing results as a first demonstration that gravitational motion in D1 is identical to that in the space-time of the General Theory. A solution of the orbit is also obtained which when approximated for the appropriate astronomical situations shows that it also satisfactorily represents gravitational motion in the Solar System. As such this solution thereby provides the proof for Statement (iii) in Section 4 above. The equation of the orbit in the axes of D0 is derived in Appendix B for comparison with that in [2].

5.1  A Planetary Orbit in the Axes of D1 - Einstein's Equation of Planetary Motion.

Note - In this Section reference is made to results obtained in Appendix C.

To derive the equation of planetary motion in the axes of D1, consider first (3.7). This equation can be integrated immediately to give

lnm = lnuws2 + k
(5.1)

Inserting the usual initial condition determines the constant of integration as

k = lnm0 u0 w0 s02
(5.2)

so that in (5.1)

muws2 = m0 u0 w0 s02
(5.3)

Substituting for m from (3.6) then gives

ws2

æ
ç
è
1
×
s
2

c2u2
 -  w2r2

c2
ö
÷
ø
1/2

 
 = 

w0 s02

æ
è
1 w02 s02

c2
ö
ø
1/2

 
(5.4)

In line with convention, this constant is now designated as h. However, from (C7) it can also be written as

h = w¢s2
(5.5)

The equation of the orbit can now be derived in the usual manner as follows. Put

s = 1

m
(5.6)

and from (5.5) and (5.6) compute the second order differential term in (C9) as

d2s

dtp2
= m2h2 d2m

df2
(5.7)

Substitution of this, together with (5.5) and (5.6) into (C9) then yields the desired equation of the orbit thus

d2m

df2
+ m = ac2

h2
+ 3am2
(5.8)

5.2  Solution of the Orbital Equation in D1

In the literature the equation of the trajectory of a planetary orbit has been obtained from an approximate solution of the orbital equation for two particular cases. Firstly, for closed orbits, it has been shown to approximate a precessing ellipse, viz. [2] pp 199, which constructs an approximate solution of (B14), (the equation of the orbit expressed in the axes of D0), for a truly elliptical orbit. A similar result is obtained in [3], pp247 Example 102, where an approximate solution of (5.8) is obtained. This solution however, applies only to a circular orbit. Secondly, an approximate solution of (B14) has been obtained for an open orbit in the extreme form of a light ray passing close to the geometrical radius of a gravitational source, viz. [2] pp 202, and has thus been shown to be a precessing hyperbola.

In this Section, an exact solution is obtained for the planetary orbit in D1, (5.8), by assuming it to be a precessing conic section and using, essentially, the method of Frobenús. This solution is then reduced to the above approximate forms via a process of logical simplification.

Note - In this Section reference is made to results obtained in Appendices B and C.

5.2.1  Equation of the Spatial Trajectory - The Basic Curve.

The equation of the basic curve of the trajectory will be derived from the first integral of (5.8) which is á priori obtained from (B8) by computing ds/dtp from the first order term in (C2), then substitution of (5.5), (5.6) and (B16) to give

æ
è
dm

df
ö
ø
2

 
c2e2

h2
- cu2

h2
- u2m2
(5.9)

Substitution of (4.7) then gives the required relationship as

æ
è
dm

df
ö
ø
2

 
c2

h2
( e2 - 1 ) +  2a  c2

h2
m - m2 + 2am3
(5.10)

the first integral of (5.8).

If the integral of (5.10) is assumed to be a precessing conic section then it will take the form

m = 1

L
{ 1 + ecos( f - W ) }
(5.11)

where

L is the semi latus rectum

e is the eccentricity of the basic curve

f is the angle of the focal point radius vector to the major axis

W is the angle of precession of an apse of the trajectory.

Differentiating (5.11) with respect to f yields

dm

df
= e

L
sin( f - W ) æ
è
1 dW

df
ö
ø
(5.12)

but from (5.11)

sin( f - W ) =  ì
í
î
1 -   æ
è
mL - 1

e
ö
ø
2

 
ü
ý
þ
1/2

 
(5.13)

Thus (5.12) and (5.13) give, after expansion

æ
è
dm

df
ö
ø
2

 
 =  ì
í
î
( e2 - 1 )

L2
 +  2m

L
- m2 ü
ý
þ
æ
è
1 dW

df
ö
ø
2

 
(5.14)

Comparing (5.14) with (5.10), to obtain the term in m3 with the correct coefficient, it is necessary to put

æ
è
1 dW

df
ö
ø
2

 
= ( b - 2am )
(5.15)

where b is some constant.

Inserting (5.15) into (5.14) and expanding gives

æ
è
dm

df
ö
ø
2

 
( e2 - 1 )

L2
b 2

L
ì
í
î
b a

L
( e2 - 1 ) ü
ý
þ
m æ
è
b 4a

L
ö
ø
  m2 + 2am3
(5.16)

Again, comparing (5.16) with (5.10), for the coefficient of m2 to be -1, it is necessary to put

b = 1 4a

L
(5.17)

so that in (5.16) this gives after reduction

æ
è
dm

df
ö
ø
2

 
( e2 - 1 )

L2
æ
è
1 4a

L
ö
ø
2

L
ì
í
î
1 a

L
( 3 + e2 ) ü
ý
þ
m - m2 + 2am3
(5.18)

Now, equating the coefficient of m in (5.18) with that in (5.10) then gives

h2
a
c2L

1 a

L
( 3 + e2 )
(5.19)

L can be determined from initial conditions applied to (5.11) to be

L 1 + e

m0
(5.20)

Inserting (5.20) and (5.4) for h into (5.19) then gives, after solving for w02

w02 = ac2( 1 + e )2m03

{ 1 - 2am0 + ( 1 + 2am0e }
(5.21)

Now, for the basic orbit to be circular, i.e. e = 0, (5.21) reduces to

w02 = ac2m03

1 - 2am0
(5.22)

For values of w0 below this, the basic orbit will be degenerative, and for values above elliptical.

For the basic orbit to be parabolic, i.e. e=1, (5.21) reduces to

w02 = 2ac2m03
(5.23)

For values of w0 below this the basic orbit will be elliptical and for values above hyperbolic.

Solving (5.21) for e gives

e 1

2am0
é
ê
ë
w02

c2m02
( 1 + 2am0 ) - 2am0
 ±  w0

cm0
ì
í
î
w02

c2m02
( 1 + 2am0 )2 - 16a2m02 ü
ý
þ
1/2

 
ù
ú
û
(5.24)

which can also be obtained by equating the constant terms in (5.10) and (5.18).

Now, for the case in which w0 is equal or very close to its terminal value in D1, its upper boundary, i.e. w0 =cm0, (5.24) reduces to

e 1

2am0
{ 1 ± ( 1 + 4am0 - 12a2m02 )1/2 }
(5.25)

and it is noted for future reference that if 4am0 << 1, (5.25) may be approximated by

e » 1

am0
(5.26)

There appears to be a second root under this criteria, i.e. at e=0. However, if the terminal value of w0 is entered into (5.22), it reduces to 3am0 = 1, which contradicts the inequality. This second root may therefore be discounted for astronomical situations.

Thus, the basic curve of the trajectory is now seen to be given by the combination of the three equations; (5.11), (5.20) and (5.24). The full solution to (5.10) is completed by the determination of the function W.

5.2.2  Precession of the Perihelion of the Basic Curve.

Determination of W is effected by substituting (5.11), (5.17) and (5.20) into (5.15) to give

æ
è
1 dW

df
ö
ø
2

 
= 1 6am0

1+e
  -  2am0 e

1+e
cos( f - W )
(5.27)

Putting

f - W = c
(5.28)

reduces (5.27) to the following elliptic integral

df = æ
è
1 6am0

1+e
2am0 e

1 + e
cosc ö
ø
-1/2

 
dc
(5.29)

the solution of which is

f æ
è
1 6am0

1 + e
ö
ø
-1/2

 
ì
í
î
æ
è
1 +  3a2m02 e2

( 1 + e - 6am0 )
ö
ø
c
am0 esinc

( 1 + e - 6am0 )
3a2m02 e2sin2c

2( 1 + e - 6am0 )
 - ¼ ü
ý
þ
(5.30)

the constant of integration being zero.

This relationship, together with (5.28) then permits the determination of W for any value of f, for any condition. Thus (5.30), together with (5.11), (5.20) and (5.24), albeit somewhat cumbersome, constitutes the exact solution of (5.10).



G1 Version 2.2.4
Ó P.G.Bass, November 2009
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