•

- , .THE DESCRIPTION OF PRECESSTh:G ELLIPTICAL ORBITS WlTH TIlE Il00KE I

NEWTO:-J Il'oVERSE SQUARE LAW.

by

M. W, Evens, H. Eckardt, R. Delatorce and R. Cheshi re,

Civil List, AlAS and UPITEC

(www,ajas,us, www,upjlec.or~. www,atQJDicpredsjon,cQw. ....ww.e!3m.o et

W\Oow.webartbiye Qrg uk)

ABSlltACT

It is demonstrated straightforwardly that a precessing elliptical orbit can be

described with a Hooke I Newton inverse square force law provided that the plane polar

coordinate system is rotating. The rotation generates a Christoffel connect ion. This is the

simplest way to describe the observed planetary orbit and is preferred by Oc kham's Razor.

Some graphical representations of the theory are given.

Keywords: EeE theory. rolal ing coordinate system, planetary orbits.

•

-- I 1~-

1. INTRODUCTION

Planetary orbits were first rationalized by Kepler in terms of three laws which

were based on the observations of Brahe. Kepler found that the orbit of the planet Mars was

an ellipse, which was later found to be precessing. For a static elliptical orbit the Kepler laws

can be explained in terms of an inverse square force law as is well known. The precessing

elliptical orbit has been considered hitherto as a non Newtonian problem, in that precession

cannot be explained with an inverse square law in a frame of reference described by the plane

polar coordinates { 1 - 12}. Einstein later claimed that his theory of general relativity can

explain the precession of the perihelion. This claim was rejected by Schwarzschild almost as

soon as it was made, and has been rejected since then. In recent papers of this series the

Einsteinian general relativity (EGR) has been refuted logically in several ways, some of them

very simple as in UFT202 (www.aias.us). So it is considered that EGR is an obsolete theory

that was always obscure.

In Section 2 it is shown by elementary methods that the precessing ellipse can

be described by an inverse square law in a rotating coordinate system ( r, r ) in which

is defi ned by the product X e . Here e is the angular coordinate of the plane polar system

( r, e ) and X is the precession constant. The rotation of the coordinate system is shown to

be described by a Christoffel connection that is antisymmetric in its lower two indices. When

the standard plane polar system ( r' e ) is used the force law needed to describe a

precessing elliptical orbit is tlle sum of an inverse square and inverse cube law. Therefore the

precessing elliptical orbit of a planet can be described entirely without Einsteinian general

relativity (EGR). Not only do we reach this conclusion, but it is also known that EGR

- 5 produces an incorrect force law, a sum of an inverse quare and inverse fourth power term in

the radial coordinate r. These are very good reasons for abandoning EGR.

2. FORCE LAWS FOR A PRECESSING ELLIPTICAL ORBIT.

The precessing elliptical orbit is defined by:

( --

where

Here x is the precession constant, l~ is the right magnitude and E is the ellipticity.

Adopt the rotating plane polar coordinate system ( r, ~ ). This is the optimal

choice or coordi nates by Ockham 's Razor, in that it rationalizes the precessing elliptical orbit

where U is the poten ti al energy . The Euler Lagrange equations are:

C~) J[ A j~ 0 --- ) -- J.j d; J( and

)1 (s) Ji tA 0 --- JF}. Jf ~ The total anguhr mPmentum is co nse rved so:

L

0 -h)

ts t 1e force. Denote· where F · 1

then:

4 -:;:.. --ct~

From Eq. ( b ):

so:

Therefore:

•

- 1 I···

u. = \ -

L ")

V\'--(

where:

\ -Therefore:

and

cl1. \ ----= ~~")

and

If:

then

where

--

\ --\- -(

r(() -

..... -

\ -~

-

The force is a Hooke I Newton inverse square law. Therefore in the rotating frame

a precessing elliptical orbit can be described without EGR at all.

This result means that the hamiltonian in the rotating frame is

\ -}

-\ !~ .

where the total linear veloc ity is: \ / . 1 . ) . ')\ ).

" "" ( ( +- r . f ) .

- ( ~ ( t - ~) - l-: ~ \ I~) - ( )€) Y\-. V\- ( )

Therefore :

Usi ng: - (J')

it is rou nd that:

whose solution is:

Th'-·rcl'ore the hami It on ian ( }3 ) desc ri bes a precess ing elliptical orbit without EGR, QED.

lf Eq. c \S) is used with the static coordinate system c r, 8 ) then:

~ ( l_ \ + .L "- - >'h () f ( () - ( ~6) Ml ') (-J ( \ . .? .

. , _ J_(\+(C•s(xe~-(l>) -- J_

(

where:

So:

and

o I,.. j' :::- mg a orcc 1'1'':

' ' :' tell is , , · . . ,J sum o I Ill vc rs , . . , , . . c squJJ c dtld tnvcrse cubed t . 1" . et ms. he EGR theor mcorrecl sum of

1 . · Y produces an

erms as shown in UFT196 . on www.mas.us.

The precise meaning of the rotaf a . ' . . tno coordmate system is deduced b . . '""'c cquarions Y conSJdenng the

X -=- < (6J~ I Y ~ o;:" f - (>s)

Consider: e -) e + :l11 - (31.)

then: - (n)

and:

)(\ ~ ({o~ (r t )lf~ ~ f ( {•If (OS ()q~- Si"-f S<~(Jrr~ '/

1

.,_ (51,._ ( ~ t Jl{0 -,.r (51'-~ {•J (:h~+ (q(~ s•~ ( )rrxJ) - c~~;

111 "hich:

) (

-- I ,,

t

It is seen th nt the coordi na te sys tem does not return to the same point after a rotation through

:l.t"{( radians. The coordinate system and frame ol' reference rotates and there is a

Christoffel connection present in the geometry. The unit vectors of the system are:

so:

lt follows that:

and: •

• .Q

Tl I. 1 .. -~

1e mear ve oc1ty ts:

,_ i_ ( oJ ~ .\:- ~Sit-. ~

"- - \ s;" ~ t l l •.s r ~(- ~--

d.~( - ~f ~p - - ~~ ~ (

•

~~

c~3) c~)

- <l~~ ( JJ: - ~ ~p - (~s) - (ltb) • - ).~~ I .u- - ~ ~ (

• • •

a result which leads to the lagrangian ( 3 ). From the equation or a precessing ellipse, eq.

( 1. ):

so:

As in UFT 196 I t le acceleration is:

~ " ~ ~ ( ~· - { ~ :>) f_{ t ( ( ~· t ) ; ~)!-.f. - (s9 Therefore: (' ~ ( ~) ~ ~·~~' f ~ ~ ~ (~~ ~ (s))

Now usc: • A\ (<"' ~ J.) (<) ~ ~ ( sJ\ ~ '%'t: ~¥ I ')

~ ~ ~*--' _(s~) a.< ~r JJ.

~ ~ ~ ~ -~ ~ ) ~ ( s•~i) = c ·s p %- . -( s~ to li nd that:

·rhcrefore· ~._ f 3

_ - (s0 <

lt follows that:

Similarly:

•• . ) r - '~

so the force is:

f - Vh -Finally use:

E (05 r to lind that:

-- I 1~

(J

-- -(

)

L

') e -(s~ L -, ) l

~ (

1 - (6~ . -

n ,_ - th.M.-6- ~( ~( )

~ {br) \ ~ ) , QED. The centrifugal forces cancel as discussed in UFT 196. which is Eq. (

fn the rotating Cr:.1me the passive rotation of axes is described by a development

ol U FT199 as:

_ s.-._(x e)]li- ~ (f.J) ' I - ( b.s ()(e) \ -- -::...

(6) ( x a) ) . ' s;'"' ( x e) ~ -

Denote: (os(xe) - s•~. ()(e) (o) -R(_

'

cos ( x e) -5;~. (x&)

then:

JQ~ --JJ) e-=-o

and the rotation generator is:

The totally antisymmetric unit tensor is:

so

f . . \ )

and the antisymmetric Christol'fel connection is:

-- I 1~

0 ::>~

\

\ X E- .. l ~

-1 - ( ~4-) 0

_ ( ts)

--

using a development ol'the method in UFT212. The precessing elliptical orbit can be

described by the Christorl'el connection ( (g ), and this is an elegant expression ofthe

philosophy or relati vity. It should replace the obsolete EGR by Ockham's Razor.

3. GRAPHICAL AND OTHER ANALYSIS OF THE RESULTS OF SECTION 2.

(Section by II. Eckardt, R. Delaforce and R. Cheshire)

THE DESCRIPTION OF PRECESSING

ELLIPTICAL ORBITS WITH THE HOOKE /

NEWTON INVERSE SQUARE LAW

M. W. Evans∗, H. Eckardt†, R. Delaforce and R. CheshireCivil List, A.I.A.S. and UPITEC

(www.webarchive.org.uk, www.aias.us,www.atomicprecision.com, www.upitec.org)

3 Graphical and other analysis of the results of

section 2

First we show that the orbits of Einstein General Relativity are di�erent fromthose of the observed kind. De�ne the radius function r(θ) of a precessingellipse:

r =α

1 + ε cos(xθ). (69)

Rewriting of this eqation gives

cos(xθ) =α

εr− 1

ε(70)

which, with aid of

sin(xθ)2 + cos(xθ)2 = 1, (71)

can be transformed into

sin(xθ)2 =2α

ε2r− α2

ε2r2− 1

ε2+ 1. (72)

This expression has to be eqated with the result of EGR for the sin(xθ)2 term:

ε2x2(2α

ε2r− α2

ε2r2− 1

ε2+ 1

)= α2

(r0a2r

+r0r3

− 1

r2+

1

b2− 1

a2

). (73)

It is seen that the left hand side is a polynom of maximum degree 1/r2 while theright hand side is one of maximum degree 1/r3. Both sides can never be equal

∗email: emyrone@aol.com†email: horsteck@aol.com

1

Figure 1: Force components F2 and F3 for di�erent values of x ≥ 1, withparameters k = 1, α = 1.

for a certain range of r. However the variable r must be able to vary because itdescribes an ellipse - reduction ad absurdum.

Next we show some examples of force laws for dependencies of 1/r2, 1/r3

and their combinations. We de�ne

F2 = −k x2

r2, (74)

F3 = −αk

(1− x2

)r3

. (75)

Both forces are shown in Fig. 1 for several values of x ≥ 1. Observe that F3is positive. The sum of both is graphed in Fig. 2. The positive contributione�ects a minimum in the total force, similar to potentials of atomic orbitals.

The equivalents of Figs. 1 and 2 for parameter values x ≤ 1 are graphed inFigs. 3 and 4. Here all force components are negative, leading to a lowering ofthe total force.

Finally we show particular forms of normal and precessing ellipses. Fig.5 presents ellipses for several ellipticities ε. In Fig. 6 the region of commoncrossing points has been enlarged. Precessing ellipses have been plotted in Figs.7-10. The factor x describes the �multiplicity� of the elllipse in case of integervalues. For non-integral values, the ellipses precess around the near integers.

2

Figure 2: Sum of force components F1 + F2 of Fig.1.

Figure 3: Force components F2 and F3 for di�erent values of x ≤ 1, withparameters k = 1, α = 1.

3

Figure 4: Sum of force components F1 + F2 of Fig.3.

Figure 5: Ellipses for several values of ε.

4

Figure 6: Enlarged view of Fig. 5.

Figure 7: Ellipses with multiplicity 2.

5

Figure 8: Ellipses with multiplicity 3.

Figure 9: Ellipses with multiplicity 4.

6

Figure 10: Ellipses with multiplicity 5.

7

ACKNOWLEDGMENTS

The British Government is thanked for the award of a Civil List Pension and the

st<t!T or AlAS and others for m<~ny interesting discussions. David Burleigh, CEO of Annexa

Jnc., is thanked for voluntary posting. Alex Hill, Robert Cheshire and Simon Clifford are

thanked for translation and broadcasting. The AlAS is established under the aegis of the

Newlands Family Trust (est. 20 12).

REFERENCES

{I} M. W. Evans, eel., "Definitive Refutations of the Einsteinian General Relativity"

(Cambridge lnternational Science Publishing, CISP, 2012 , WV/'vv.cisp-publishing.com)

[2] M. W. Evans, ed . .1. Found. Phys. Chem., (CTSP, 2011 onwards).

{3} M. W. Evans, S. Crothers. 1-1. Eckardt and K. Pendergast, "Definitive Refutations ofthe

Einstein Field Equation'' (C1SP. 201 1).

[4) K. Pendergast, ··The Life of" Myron Evans" (CISP, 2011).

{ 5} M. W. Evans, H. Eckardt and D. W. Lindstrom, "Generally Covariant Unified Field

Theory" (Abram is, 2005 - 201 1) in seven volumes.

( 6: L. Felker. ··The Evans Equations of Unified Field Theory" (Abramis 2007, translated into

Spanish by Alex Hill on WW\·v.,aias.us).

f7} M .W. Evans and S. Kielich. eels., "Modern Nonlinear Optics" (Wiley, 1992, 1993, 1997,

2001 ). in six volumes and two edit ions.

{8} M .W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the B(3)

Field" (World Scientific 2001).

:9: M. W. Evans and .1.- P. Vigier. ·'The Enigmatic Photon'' (Kiuwer, Dordrecht, 1994 to

2002, hardback and softback) in ten volumes.

{ 1 0} M. W. Evans and A. A. 1-lasanein, "The Photonagneton in Quantum Field Theory"

(World Scientific 199-+)