IL NUOV0 CIMENTO

VOL. 7 C, N. 1

Gennaio-Fehbraio 1984

Perihelic Shift of Planets Due to the Gravitational Field of the Charged Sun. M . T . T E H a n d D. I~ALASKA~ Department o] Physics, Marathwada University - A~trangabad 431004, rndia

(ricevuto il 22 Agosto 1983)

Summary. - - The perihelic shift of planets due to the charged Sun is calculated. The results when compared with experimental shifts suggest that the planets Mercury, Venus and Icarus do not possess self-electromagnetic fields. PACS. 95.10. - Fundamental astronomy.

REIss~m~ a n d NO~])STRO~ o b t a i n e d t h e following line e l e m e n t b y introducing t h e charged m a s s p o i n t in v a c u u m (1): (1)

ds~(1 =

2raCe'' ~2r~

(

1

2m

Ce~o ~-1 2c~r2 ] dr ~"- - r2(dO ~ + sin ~0 d~ ~) ,

where m ~ k M / c ~, C ~ - - 8 n k / c 2 a n d % is t h e charge of a g r a v i t a t i n g body, for e x a m p l e , t h e Sun; M is t h e m a s s of t h e Sun. W e wish to s t u d y t h e m o t i o n of a p l a n e t in t h e field due to t h e charged Sun.

1. - P l a n e t a r y

motion.

The m o t i o n of a p l a n e t in c h a r g e d Sun's g r a v i t a t i o n a l field is o b t a i n e d b y solving t h e geodesic equations o b t a i n e d f r o m the line e l e m e n t (1). W e

(1) n. ADLER, M. BAZIN and M. SCHIFFER: Introduotior~ to General Relativity (New York, N.Y., 1965), p. 401. 130

PERIttELIC

SHIFT

OF PLANETS

DUE

TO THE

GRkVITATIONA.L

FIELD

ETC.

131

consider the motion in t h e 0 ~ n/2 plane. The geodesic equations t h e n are d

(2)

dss (r20") -~ r2 sinO cosO~ '~ ,

(3)

d ds (r~ sin2 0 ~') = 0,

d i(1 2or

(4)

d-~

2~r~

t

=o.

The geodesic equation for r is obtained b y dividing the line element (1) by ds ~ (5)

1=~1 \

2m

Ce~)~

r

2e2r~/

~ \1

C2t~

2m r

eel) I-1 2e~r ~]

r ' 2 - - r2~ 92

To simplify (5) further, we first consider the integral of eq. (4)

(6)

(1

r

2m r

2c~r2/t"

•

1,

where 1 is the constant of integration. Similarly eq. (3) integrates to (7)

r~'=

h

and here h is the constant of integration. Now we take r as a function of ~ instead of s and denote the differentiation ~o.r.t.q~ by a prime, so t h a t h r" = 7~ r ' .

(s)

Substituting eqs. (6)-(8) in eq. (5), we obtain (9)

2m r

1

Ce~ 2C27 "2

--

~12__rI2

b y changing the variable from r to u = ] / r eq. (9) takes the form

(10)

e21~_ 1 u'~

--

-

-

h~

+~u

§ ~ - - 1 u~+2mua ~ - a u 4,

~vhere (11)

a

--

r 2~2

4=ke5 ~,4

132

~ . T. T~LI a n d D. PALASKAR

Differentiating e q u a t i o n (10) a~.r.t.q~, we obtain t h e e q u a t i o n of the orbit in the form (12)

u" ~-

(1a- -)~

o ~- 3 m u 2 ~-

u -~ ~

2au

3 .

This is t h e differential e q u a t i o n for t h e v a r i a b l e r.

3. -

Perihelie

shift.

E q u a t i o n (12) can be w r i t t e n as

(13)

u" + o ~ u : -

U2 + - ~

A §

U$ ,

where m

(1~)

a

A

h2 ,

A ~

s

= 3mA ,

#

h2 ,

---- 2 a A ' ,

a

co2=-- 1 - - - h2 a n d # are v e r y small quantities. W e assume t h e solution of eq. (13) of t h e f o r m (15)

u ----Uo -t- ev -[- ~' W ,

where (16)

A Uo~-- ~ + B cos w~

is t h e general elliptical solution of t h e ~ e w t o n i a n equation

(17)

u~ + ~Uo = A.

B y following t h e t e c h n i q u e of Adler et al. (2) t h e solution of eq. (13) is ob-

R. AI)L~R, M. BAzIN and M. SCIIIFFER:Introduction to General Relativity (New York, N . Y . , 1965), p. 185, 186.

(2)

FERIHELIC

S H I F T OF P L A N E T S DU}] TO T H E

GRA,V i T A T I O N A L F I E L D

133

ETC.

tained as u =

(18)

I[.A +

A+

+ 3A.,ll + [ eB2

e"AB2\

+ B cos w(9 - - b~) - - ~

'

+ 2--A~w2) cos 2wq~

Bae ~

32~2A, cos 3 w ~ ,

where

e

(19)

e' [ 3A-"

B2)

b = o,--z + ~ X~ ; o ,, ~ ~-~, .

F r o m (18) we see how t h e small quantities e a n d e' introduce, i) pert u r b a t i o n s to the c o n s t a n t A, ii) periodic p e r t u r b a t i o n t e r m s (3) and (4) in eq. (18). I n addition to these p e r t u r b a t i o n s , the phase is also p e r t u r b e d by e and e' t h r o u g h the factor b. Thus the phase cot --> (o9(1 - - b) and this changes the position of the perihelion of the orbit, i.e. the planet s t a r t i n g from the positioll r = 0 does not r e t u r n to the same position after ~o~ = 2~ or ~o~ = = 2~n. Thus after each revolution the perihelic position goes on changing. The m a x i m u m value of u occurs when (20)

(o~(1 - - b) = 2 ~ n ,

2~n ~ := - - (1 ~- b).

or

(0

Thus successive perihelia occur at intervals of (21)

A o ~ = 2~(1 + b)

instead of 2~ as in periodic motion. The perihelic shift per revolution is given by (22)

~

= 2:tb.

S u b s t i t u t i n g eq. (19) in eq. (22), we have

s u b s t i t u t i n g the values of e, e', A, A' and eo2 from eq. (14) into eq. (23) we get

(24)

~

2 [3k2M2 6k2M~a 15k~M~a2c~" aB2 a~B-2c21 :: ~ / ~ - ~ + -~.-- + 2H6 + ~h~ + T ~ J"

The perihelic shift per c e n t u r y is given by (25)

s --

T

'

where T is the period of revolution expressed in units of century.

134

~I. T. T:ELI and D. I'ALASKAI~

I f r is t h e m e a n d i s t a n c e of t h o p l a n e t f r o m t h e S u n , we i n t e g r a t e t h e r e l a tion (iT H = r" d-t

(26)

over one revolution and approximately

obtain

2~tr ~-= H I ' .

(27)

We also have Kepler's third law T" -- = d, ra

(28)

w h e r e d h a s t h e s a m e v a l u e for a l l p l a n e t s . eq. (24) b e c o m e s

(29)

[3k~.~V~

s - - 2~ [ c2(2~t) ~ r -t -f-

T h u s w i t h eqs. (25), (27) a n d (28)

6 k ~ ~ a ~ r-~ + 15k~ M'Za~c~di r 4 + (2~) ~

2(2~) n

-'7

aB 2

4V~

aO.B2c2%/~

r-~ +

8(2~)~

r-~

}

"

:Now B cos ~o~v --- uo - - A/eo 2 a n d w e c a n t a k e B "~ uo = 1/r for a n e a r l y c i r c u l a r o r b i t . E q u a t i o n (29) t h e n b e c o m e s

{ 3k''M'z~/~ (30)

s -- - 2~ - c ~

6k2M2ad~ r-i r - t + - (2Jr)'

:15k2M"a~ -'7

r-I

2(2zt) s

4V~ r - ~

~

-'7 - -

(31)

[3k~M'~/~

r-,j

~ (1 + 24k'M~a~] ~_~+ (2~)' !

8 - 2 ~ / c~(2~)2-r-g+~

+ 4(2~)2\ +

~

!r-~}

b y s u b s t i t u t i n g t h e v a l u e of a f r o m eq. (:1:1 i n t o eq. (31)

(32)

s :

2z[

[3k2~/~ c*~

r-t--

~ke~ 1 -4- 24k2M2d2~r - ] - -

~

(~+

(2~), !~'} 9

T h e f i r s t t e r m i n eq. (32) is E i n s t e i n ' s t e r m f o r t h e p e r i h e l i e s h i f t a n d t h i s v a r i e s as t h e - - 5 / 2 p o w e r of t h e d i s t a n c e of t h e p l a n e t f r o m t h e S u n .

PERIHELIC

SHIFT O F P L A N E T S

DUE

TO THE

GRAVITATIONAL

F I E L D ETC.

135

We can define the effective charge of the Sun as

(33)

eS=eo i+

24k2M2d2~89

/"

The second a n d t h i r d terms in eq. (32) are, therefore, the effective charge contributions to the Einstein's t e r m which v a r y , respectively, as the - - 7/2 and - - 9/2 power of t h e distance of t h e p l a n e t f r o m t h e Sun. The second and t h i r d t e r m s m a y be i n t e r p r e t e d as first-order and second-order p e r t u r b a t i o n s due to a g r a v i t a t i n g body. DICKE predicted the perihelic shift (a)

3kMBd i SD - - - r-~

(34)

(2~)3

by considering a deformation in the solar shape and the shift obtained varies as the--7/2 power of the distance of the planet from the Sun. From the second term of eq. (32) and eq. (34) we can then conclude that the deformation in the solar shape is equivalently described by the gravitational field due to the charge of the Sun. We can express s up to the second order of a*/r as (

a*

s---- sEin 1 - - - - - t - - -

(35)

a*~

9

Since experiments measure s which includes contributions of all kinds such as solar mass gravity, charge g r a v i t y , etc. We can take t h e n s ---- s~,. Considering eq. (35) to first order in a*/r, we can write for a* (36)

a* --~ r sEi~--s~, 8Ein

I n t h e following table we give the theoretical and the observed precession of the p l a n e t a r y orbits (4). TABLE I. Planet

Distance from the Sun, r (cm)

General-relativity shift s (s/century)

Observed shift s (s/century)

Mercury Venus Earth Icarus

58.1011 108.1011 149.1011 161.1011

43.03 8.6 3.8 10.3

43.11 =]=0.45 8.4 =L4.8 5.0 =~ 1.2 9.8 =[=0.8

(a) R. ADLER, i~. BAZIN and M. SCHIFFER: Introduction to General Relativity (New York, N. Y., 1965), p. 204. (4) S. WEINBERG: Gravitation and Cosmology: Principles and Application o/the General Theory o/ Relativity (New York, N.Y., 1972), p. 198.

136

,~. T. TELI a n d D. PALASKAR

As the Einstein predicted shift and the observed shifts agree, the difference is v e r y small for Mercury, Venus and Icarus. This can lead us to the conclusion t h a t the orbits of Mercury, Venus and I c a r u s are nearly unaffected b y the gravitational field due to the charge of the Sun. The e x p e r i m e n t a l value for the E a r t h is greater t h a n Einstein's value and, when charge g r a v i t y is considered, the theoretical value which becomes smaller t h a n Einstein's value again departs more from the experimental one. Thi.~ means t h a t there m a y be some anomalous t e r m to enhance the theoretical value to the experimental value of the perihelic shift for the E a r t h .

9

RIASSUNTO

(*)

Si ealcola lo spostamento perielieo dei pianeti dovuto al Sole carico. I risultati, se confrontati con gli spostamenti sperimentali, suggeriscono ('he i pianeti Mercurio, Venere e Iearo non hanno eampi elettromagnetici propri. (*)

T r a d u z i o ~ e a eura della R e d a z i o n e .

Career n e p s r e a ~

nJxaHeT, o6yc~oa~eHm~fi FpaBHTalIHOHHIdM no~eM 3apm~etmoro Comma.

Pe3mMe(*). - - B~l~_ttc~aerc~ c2(arir n e p n r e a r m n z a n e r , o S y c a o s a e H n b n ~ 3 a p a ~ e H n I , iM Co~atleM. [ I o n y q e r i r ~ i e pe3ynbraTJ, I n p ~ cpaaHen24~t c a r c ~ e p n M e r r r a n b r m h v m cgattraMH n p e g ~ o ~ a r a i o T , ~To n a a n e r b t M e p r y p r r t i , B e n e p a n H r a p n e rrMemT c o 6 c T a e m m r x 3~e~TpOMarHHTHbIX iioylel].

(') l-Iepeaet)eno pee)arque~.

~) by Societ~ Itallana di Fisica Propriet~ letteraria risorvata Direttore responsabfle:

RENATO AI~GELO RICCI

Stampato in Bologna dalla Tipografla Compositori cot tipi della Tipografla Monograf Questo fascicolo $ stato licenziato dai torchi il 19-VI-1984

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