Content uploaded by Eduardo D. Greaves

Author content

All content in this area was uploaded by Eduardo D. Greaves on Oct 22, 2013

Content may be subject to copyright.

1

NASA’s astonishing evidence that c is not constant: The pioneer anomaly

E. D. Greaves

Universidad Simón Bolívar, Laboratorio de Física Nuclear, Apartado 89000, Caracas

1080 A, Venezuela. E-mail: egreaves20002000@yahoo.com

For over 20 years NASA has struggled to find an explanation to the

Pioneer anomaly. Now it becomes clear the solution to the riddle is that

they have uncovered evidence that c, the speed of light, is not quite a

universal constant. Using J. C. Cure’s hypothesis that the index of

refraction is a function of the gravitational energy density of space and

straightforward Newtonian mechanics, NASA’s measurements provide

compelling evidence that the speed of light depends on the inverse of the

square root of the gravitational energy density of space. The magnitude of

the Pioneer anomalous acceleration leads to the value of the primordial

energy density of space due to faraway stars and galaxies: 1.0838. x 10

15

Joule/m

3

. A value which almost miraculously coincides with the same

quantity: 1.09429 x 10

15

Joule/m

3

derived by J. C. Cure from a

completely different phenomenon: the bending of starlight during solar

eclipses.

PACS numbers: 95.55.Pe, 06.20.Jr, 04.80.Cc, 95.10.-a

Introduction

Anderson and collaborators at the Jet Propulsion Laboratory (JPL) have reported [1] an

apparent, weak, long range anomalous acceleration of the Pioneer 10 and 11 with

supporting data from Galileo, and Ulysses spacecraft. [2, 3] Careful analysis of the Doppler

signals from both spacecraft have shown the presence of an unmodeled acceleration

towards the sun. By 1998 it was concluded from the analysis, that the unmodeled

acceleration towards the Sun was (8.09 +/- 0.20) x 10

-10

m/s

2

for Pioneer 10 and of (8.56

+/- 0.15) x 10

-10

m/s

2

for Pioneer 11. In a search for an explanation, the motions of two

other spacecraft were analyzed: Galileo in its Earth-Jupiter mission phase and Ulysses in a

Jupiter-perihelion cruise out of the plane of the ecliptic. It was concluded that Ulysses was

subjected to an unmodeled acceleration towards the Sun of (12 +/- 3) x 10

-10

m/s

2

. To

investigate this, an independent analysis was performed of the raw data using the

Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP),

which was developed independently of JPL. The CHASMP analysis of Pioneer 10 data also

showed an unmodeled acceleration in a direction along the radial toward the Sun. The value

is (8.65 +/- 0.03) x 10

-10

m/s

2

, agreeing with JPL’s result. Aerospace’s analysis of Galileo

Doppler data resulted in a determination for an unmodeled acceleration in a direction along

the radial toward the Sun of, (8 +/- 3) x 10

-10

m/s

2

, a value similar to that from Pioneer 10.

All attempts at explanation of the unmodeled acceleration as the result of hardware or

software problems at the spacecraft or at the tracking stations have failed. A very detailed

description of the Pioneer anomaly, of the measurements and of the analysis was given by

the JPL team [4]. Two conferences have been carried out on the subject, in 2004 [5] and in

2005 [6] and although several explanations have been advanced, no clear consensus exists

of the cause of the weak [7] anomalous acceleration experienced by the various spacecraft.

With no plausible explanation so far, the possibility that the origin of the anomalous signal

2

is new physics has arisen.[8] Very recently evidence of the puzzling phenomenon was

found in the motion of other spacecraft. [9]

The Pioneer anomalous acceleration a is derived from the Doppler drift fΔ of the base

frequency

o

f detected:

(

)

c

a

ff

o

=Δ In this paper the anomalous drift is shown to be due to

a change of the index of refraction of vacuum, a function of the gravitational energy density

of space predicted by the Curé hypothesis [10]. It affects

c the speed of light in space far

from the influence of the sun.

1.- Energy density of space.

By energy density of space we mean the classical energy density (Energy per unit volume)

associated with the potential energy of all forms of force: electric, magnetic, gravitational

or any other force in nature. In particular, to be associated to gravitational energy of nearby

massive bodies such as the Sun and the Earth which we can readily calculate, and to the

gravitational energy density produced by the gravitational field of the stars and far away

galaxies, not so easily estimated.

The energy density of space associated with the presence of static electric

E

and magnetic

B

fields are given by:

22

2

1

2

1

BE

o

o

μ

ερ

+= (1)

Where

0

ε

and

0

μ

are the electric permittivity and the magnetic permeability of space

respectively. The equivalent energy density associated with a gravitational field

g

(m/s

2

) is

given by

G

g

π

ρ

42

1

2

=

(2)

with

G

the Universal constant of gravitation. Hence any volume of space is immersed in

the universal primordial field of energy

*

ρ

which includes the immediate gravitational field

due to the presence of our own galaxy superimposed on the energy fields of all far-away

galaxies. Thus the energy density in the surface of the Earth and in the proximity of the Sun

is given by:

ES

ρρρρ

++=

*

(3)

where the energy density due to the Sun

S

ρ

produced by the gravitational effect of the

mass of the Sun

S

M is obtained from (2) with

2

/ rGMg

S

=

4

2

8 r

GM

s

S

π

ρ

=

(4)

Here

r

is the distance from the centre of the Sun to the point in question. And

E

ρ

is the

energy density due to the gravitational effect produced by the mass of the Earth and is

calculated in analogous manner. The acceleration of gravity

S

g

due to the Sun at the radius

of the Earth’s orbit is

S

g

= 0.00593 m s

-2

. Hence the Sun’s energy density at the Earth orbit

is

S

ρ

= 2.097 x 10

4

Joules/m

3

. With the Earth’s acceleration of gravity the energy density

due to the Earth at the surface is

E

ρ

= 5.726 10

+10

Joules/m

3

and the universal primordial

3

energy density

*

ρ

is estimated [10] at 1.09429 x 10

15

Joules/m

3

.

This is a value arrived at by

an analysis of the deflection of light by the Sun’s energy field considered as a refraction

phenomenon as reviewed below. [11]

J.C. Curé [10, p. 276] explains the energy density of space in the following illuminating

words:

“Every celestial body is surrounded by an invisible envelope of gravitostatic

energy caused by the matter of the body and given by Eq. (104). (Our Eq. 4) To

proceed with a colorful description, let us assign a yellow color to the sun’s

gravitostatic energy. Let us picture the background cosmic energy with a bluish

color. Now we can see, in our imagination, that the sun is surrounded with a

green atmosphere of energy. The green color fades away into a bluish color as

we recede from the sun.”

2.- Effect of energy density of space

Now let us consider the hypothesis that the speed of light is a function of the energy density

of space

ρ

which in the neighbourhood of the sun is determined by a constant background

value due to the distant galaxies plus a smaller value due to the gravitational presence of the

Sun’s mass as seen by (3) above.

We assume the speed is inversely proportional to the square root of the energy density by

the use of the Curé hypothesis [10, p 173] given by relation (5):

ES

k

c

ρρρ

++

=

*

'

(5)

This implies that the speed of light decreases near the Sun and increases far away from the

sun. We may then assign an index of refraction

n to space such that n = 1 in vacuum space

near the Earth, as we usually do, and assign an index

'n

< 1 far away from the Sun, in deep

space, where the speed of light

'c is greater and is given by:

'

'

n

c

c = (6)

so that the index of refraction there is

'/' ccn

=

.

Using (5) we may write expressions for c and 'c and obtain the index of refraction, 'n , far

away from the Sun in terms of

AUS1

ρ

the energy density of the Sun at the distance of the

Earth: one Astronomical Unit (

r

= 1 AU),

E

ρ

the energy density of the Earth at the

surface,

Sfar

ρ

, the energy density of the Sun, relatively far away but in the vicinity of the

Sun and

*

ρ

the interstellar primordial energy density in the vicinity of the Sun [12] as:

EAUS

EfarSfar

n

ρρρ

ρρρ

++

++

=

1

*

*

'

(7)

Strictly speaking, relation (7) should contain in the numerator and denominator the

gravitational energy density due to all the other planets. However, the contribution is

negligible due to the

4

/1 r factor in the energy density, unless 'n is being calculated near a

planet.

At this point it is fitting to address the order of magnitude of the quantities being discussed.

With

n = 1 at the Earth at 1 AU from the Sun, the index of refraction 'n further away from

the Sun is dependent on the relative magnitudes of the energy density values that enter into

4

Eq. (7), i.e. the relative value of the Sun’s energy density, the Earth’s energy density and

the primordial energy density

*

ρ

of space due to the stars and far-away galaxies.

If we plot relation (4) we find that

S

ρ

falls of rapidly as we go away from the Sun, see Fig.

1, and it becomes negligible for distances of say

r

> 10 AU compared to the universal

primordial energy density estimated by Curé [10, p 279] at 1.094291 x 10

15

Joules m

-3

.

Entering these values into (7) we find that

'n is smaller than one for

r

> 1 AU, and it is

also smaller than one for

r

< 1 AU due to the energy density of the Earth which, near the

surface, is much greater that the sun’s energy density. But the numerical value of

'n

is very

nearly one, differing only by a very small amount (see Table I). Hence the values of the

speed of light calculated at different distances from the Sun changes little from the

accurately measured value on the surface of the Earth at a distance of 1 AU from the Sun.

These minute changes in the speed of light or of the index of refraction of space are

consistent with the tiny magnitudes of the accelerations reported by the Pioneer anomaly.

With our knowledge of the energy density of the Sun and Earth, relation (7) for the index of

refraction

'n may be used to determine the primordial energy density of space,

*

ρ

, if we do

an independent measurement of the index of refraction of space,

'n , far away from the Sun.

Solving for

*

ρ

we get:

(

)

1'

'

*

2

1

2

−

+−+

=

n

n

EAUSEfarSfar

ρρρρ

ρ

(8)

In this relation

'n is the index of refraction at the distance where

)(

EfarSfar

ρ

ρ

+

is

calculated.

0.00.51.01.52.02.5

10

-9

10

-3

10

3

10

9

10

15

Energy Density (J m

-3

)

Heliocentric distance (AU)

Figure 1. Energy density of space as a function of distance from the sun. Top line,

energy density due to the stars. Middle line, Sun's gravity + Earth. Bottom line,

energy density due to Earth. (Along a radial line Sun–Earth)

3.- Doppler Effect.

5

The frequencies of signals received from spacecraft are affected by their movement through

the Doppler Effect. In fact the first order Doppler Effect is normally used to determine the

speed of distant spacecraft. An accurate oscillator “clock” on board emits a signal in the

form of an electromagnetic wave at a base frequency

o

f . If the spacecraft moves at a

velocity,

v, relative to the receiving station the frequency f of the clock as perceived by

the receiver is shifted from

o

f by an amount f

Δ

:

⎟

⎠

⎞

⎜

⎝

⎛

=−=Δ

c

v

ffff

oo

(9)

Hence

⎟

⎠

⎞

⎜

⎝

⎛

−=

⎟

⎠

⎞

⎜

⎝

⎛

−=

c

v

f

c

v

fff

ooo

1

(10)

This is the frequency received when v is in the direction away from the receiver, i. e. the

signal of a receding spacecraft is Doppler-shifted towards lower frequencies (red-shifted).

The reverse occurs if the spacecraft moves towards the receiver, in which case the received

signal is Doppler-shifted towards higher frequencies (blue-shifted).

Above we assumed a “clock” on board for clarity in the argument. However, in the case of

the Pioneer spacecrafts this is not true. The signals transmitted by the Pioneer spacecrafts

are re-transmission of Earth-sent signals. Assume the frequency transmitted from Earth

is

e

f , the spacecraft is in motion relative to Earth hence the frequency of the signal received

at the spacecraft for retransmission is not

e

f but rather a Doppler shifted frequency

o

f . The

shift is given by a relation analogous to (9): In the spacecraft frame of reference Earth is

receding with speed v. Hence the signal received is Doppler shifted by an amount

s

fΔ

⎟

⎠

⎞

⎜

⎝

⎛

=−=Δ

c

v

ffff

eoes

Solving for

o

f we obtain a relation like (10):

⎟

⎠

⎞

⎜

⎝

⎛

−=

c

v

ff

eo

1

Which substituted in (10) gives:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+−=

⎟

⎠

⎞

⎜

⎝

⎛

−=

2

2

2

2

11

c

v

c

v

f

c

v

ff

ee

Neglecting the second order term the Doppler-shifted frequency

f received on Earth due to

the spacecraft in motion with speed v is

⎟

⎠

⎞

⎜

⎝

⎛

−=

c

v

ff

e

2

1

and the change relative to the Earth-sent frequency is:

⎟

⎠

⎞

⎜

⎝

⎛

=−=Δ

c

v

ffff

ee

2

(11)

4.- Effect of Gravity on speed of spacecraft

A spacecraft that is receding into deep space away from the Sun does not move with a

constant velocity. This is because it is affected by the gravitational attraction of the Sun.

6

The effect is that the receding spacecraft is affected by a change of speed towards the Sun

which is equal to the magnitude of the Sun’s acceleration of gravity at the position of the

spacecraft. The acceleration is in the direction of the Sun which is approximately in a

direction opposite to its receding speed.

For a deep space probe spacecraft the acceleration

a is given from Newton’s second law by

mFa /= with

F

the gravitational force of the Sun on the spacecraft and m the spacecraft

mass.

F

is given by Newton’s relation:

2

/ rmGMF

S

=

with

G

the universal constant of

gravitation, 6.67300 × 10

-11

m

3

kg

-1

s

-2

, and

S

M the Sun’s mass, 1.98892 × 10

30

Kg, hence

the acceleration of the spacecraft is:

(12)

where

S

r is the radial distance from the spacecraft to the centre of the Sun.

The speed of the spacecraft is time dependent and is given by:

atvv

−

=

0

with

0

v the

speed at some time

0=t , and a the acceleration given by (12):

t

r

GM

vv

s

s

o

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−=

2

(13a)

If we wish to take into account the gravitational force of the Earth we must include a term

similar to (12):

t

r

GM

t

r

GM

vv

e

e

s

s

o

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−=

22

(13b)

Where

e

M is the mass of the Earth 5.98 x 10

24

Kg and

e

r is the distance to the spacecraft

from the centre of the Earth.

5.- Doppler effect with c affected by the energy density of space

Let us now consider a Pioneer spacecraft far in deep space, in a region of space where 'n <

1 re-transmitting an Earth-sent base frequency

e

f and moving away from a receiver station

at a hypothetical steady (constant) velocity

v.

The frequency

f and the frequency shift f

Δ

of the signal perceived by a receiver will not

be given by relation (11) above but rather by:

⎟

⎠

⎞

⎜

⎝

⎛

′

=

′

−=

′

Δ

c

v

ffff

ee

2

(14)

The primed variables are the values affected by the fact that the speed of light

'c in the

region of the spacecraft is different.

Substituting

ncc

′

= /' we get:

n

c

v

ffff

ee

′

⎟

⎠

⎞

⎜

⎝

⎛

=

′

−=

′

Δ

2

(15)

The meaning of Eq. (15) is that the frequency shift perceived at the receiving station is

smaller because

'n < 1. Accordingly it would correspond to a smaller Doppler shift and

hence interpreted by an observer, unaware of the value of

'n , as due to a receding velocity

of the spacecraft that is smaller that it actually is.

2

s

s

r

GM

a

=

7

Now let us consider the effect on the Doppler signals on a spacecraft whose speed is

affected by the gravitational attraction of the Sun and the Earth. The speed in not constant

but rather a function of time given by Eq. (13) above,

hence it is Doppler shifted by:

'

)(

2'

22

n

c

r

tGM

r

tGM

v

ff

e

e

s

S

o

e

−−

=Δ

With

f

′

Δ

being a function of time, the time rate of change of the Doppler shifted signal is:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+−=

Δ

22

'2

'

e

e

s

Se

r

M

r

M

c

Gnf

dt

fd

(16a)

However, if we neglect the change of the speed of light due to the energy density of space

we would have the previous relation with

'n = 1 as follows:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+−=

Δ

22

2

e

e

s

Se

r

M

r

M

c

Gf

dt

fd

(16b)

Hence the “Excess” Doppler shift

D

E (Hz/s) due to the effect of the energy density of space

is given by the difference between these two relations:

dt

fd

dt

fd

E

D

Δ

−

Δ

=

'

Or

)'1(

2

22

n

r

M

r

M

c

Gf

E

e

e

s

Se

D

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+= (Hz/s) (17)

Relation (17) gives the “Excess“ Doppler signal that is detected by a receiving station on

Earth and interpreted as an anomalous acceleration towards the Sun due to the effect on the

Doppler frequency by the higher speed of light in the interstellar medium as compared with

the speed of light,

c, on Earth.

Upon examination of Eq. (17) we see that the term in the parenthesis,

)'1( n− , is very small

owing to the fact that

'n

is smaller than one, but very near to one. At a distance of 20 AU

from the sun this term is equal to 0.0000572. The term on the right of Eq. (17), excluding

)'1(

n− , is the factor )/2( cf

e

times the gravitational acceleration of the Sun and the Earth

at the distance

r

, i.e. it is the drift of the Doppler signal due to the gravitational

acceleration at that point. An acceleration which is mainly due to the Sun.

The Pioneer anomaly reported as a weak acceleration,

a, toward the Sun is calculated from

the time rate of change of the Doppler shift, Eq. (11):

a

c

f

dt

dv

c

f

dt

fd

E

ee

D

22

==

Δ

=

(Hz/s)

Hence the anomalous acceleration is:

)'1(

22

n

r

M

r

M

Ga

e

e

s

S

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+=

(m/s

2

) (18)

With

'n given by Eq. (7).

Examination of (18) and (7) shows that the only unknown parameter is

*

ρ

, the primordial

energy density of space due to the stars and far-away galaxies. Hence we are able to predict

the magnitude of the Pioneer anomaly with

*

ρ

as a single adjustable parameter.

8

We may use Eq. (18) in several ways:

i.- With the empirical value of the “Excess” Doppler shift,

D

E , measured by Anderson and

collaborators we can calculate what is the index of refraction

'n for a particular position of

the deep space probes. Solving Eq. (17) for

'n

:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

−=

22

2

1'

e

e

s

s

o

D

r

M

r

M

Gf

cE

n

(19)

This then allows determination of the speed of light

'c in that position with ncc

′

= /' . It also

allows calculation of the energy density of space

*

ρ

due to the primordial energy field with

relation (8) assuming the Curé hypothesis given by relation (5).

ii.- The second way we can use Eq. (18) is to calculate independently the values of the

unmodeled acceleration as a function of distance from the Sun which is what is reported

[1,4]. Eq. (18) may be written in terms of the true acceleration of gravity

g

a

, Eq. (12), as:

)1(

2

n

c

af

E

ge

D

′

−=

Hence the “Excess” Doppler signal detected (Hz/s) is due to a fictitious “Excess”

acceleration

E

a

given by the real acceleration of gravity

g

a (m/s

2

) times the factor )'1( n

−

,

i.e.

()

naE

ga

′

−= 1

The factor

)'1( n−

is due to the variation of the index of refraction of space, or the change

of the speed of light due to the energy density of space.

We wish to calculate this expression for the “Excess” acceleration

a

E as a function of the

distance from Earth. We take into account only the effect of the Sun, due to its large mass,

and of the Earth due to its large magnitude in its proximity, and neglect the effect of all the

other planets. Using (7) and (12) in the previous relation the “Excess” acceleration is given

by:

⎟

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+=

4

2

4

1

2

*

4

2

4

2

*

22

8

8

1

e

e

AUS

s

ex

e

sx

s

ex

e

sx

s

a

r

M

r

M

G

r

M

r

M

G

r

M

r

M

GE

π

ρ

π

ρ

(20)

Where

Sx

r ,

ex

r are the distances from the centre of the Sun and Earth to the

x

position of

the spacecraft,

AUS

r

1

and

e

r are the distances to the surface of the Earth, and

S

M ,

e

M are

the masses of the Sun and Earth respectively.

6.- Results

Here we show the numerical results of calculations using the theory above.

9

i.- With the use of (18) and of data published [Ref. 4, p 15] of the frequency used in the

transmission to the pioneer spacecraft of

e

f = 2295 MHz and the “Excess” Doppler shift,

D

E , a steady frequency drift of (5.99 ± 0.01) × 10

−9

Hz/s from the Pioneer 10 spacecraft [

4, p 20] we calculate that the index of refraction

'n at 20 AU from the Sun is:

'n = 0.9999735679

†

(21)

With this value the accepted speed of light measured on the Earth at 1 AU as c =

299792458 m/s becomes at 20 AU the slightly higher value of

'c = 299800382 m/s .

The value (21) is the result of an empirical measurement of the index of refraction of space

at 20 AU by NASA’s careful measurements of Pioneer signals.

With this value and the use of Eq. (8) we can calculate the primordial energy density of

space

*

ρ

, using the Sun’s and the Earth’s energy density at 1AU and at 20 AU. The value

calculated is:

*

ρ

= 1.0838. x 10

15

Joule/m

3

. (22)

This value coincides with the value of

*

ρ

= 1.09429 x 10

15

Joule/m

3

calculated by Curé on

the basis of an entirely different phenomenon: The bending of starlight rays by the

gravitational field of the Sun. We outline here the calculation done by Jorge Céspedes-Curé

[10, p. 279]. It consists of using the hypothesis of Eq. (5) interpreted as a change of the

index of refraction of space, and using the analysis carried out by Prof. P. Merat [13] in

1974 [10, p 274] for 297 starlight deflections measured in 9 observations of 6 solar

eclipses. With the results of Merat’s analysis of the astronomical observations of the

bending of starlight rays by the gravitational field of the Sun, Curé determines the energy

density of space.

ii.- The result of calculating the Pioneer anomaly predicted by (20) as a function of

distance is shown in Fig. 2. To construct this curve the space energy density

*

ρ

was used

as an adjustable parameter. The value chosen

*

ρ

= 0.25 x 10

15

Joules/m

3

gives a better fit

around 30 AU to the experimentally measured anomalous acceleration as a function of

heliocentric distance.[14]

7.- Discussion.

The measurements of the Pioneer anomaly are not very precise. They are of the same order

of magnitude of the errors in the measurements and with this imprecision they do not show

a clear variation with the distance to the sun. However, considering the wildly different,

magnitudes of the data that enter the relations used to calculate the energy density given by

(22) (Gravitational constant, mass of the Sun and Earth, both masses squared, speed of

light, distance of Sun and Earth to spacecraft squared, both distances to the forth power,

frequency and frequency drift of the Pioneer transmissions) it seems miraculous that the

calculation of the energy density

*

ρ

in deep space differs by less than 1 % of the value

predicted by Curé on the basis of a completely different phenomenon: starlight deflection

by the Sun.

†

To 10 digits, although rightmost digits are not significant due to imprecision of E

D

10

Figure 2. The Pioneer anomalous acceleration predicted with the theory as a function of the

distance along a Sun-Earth radial line. (Radius of Earth = 4.26E-05 AU) The experimental

data shown, obtained by NASA, was derived from Fig. 7 of Ref. [4].

The puzzling fact that the anomalous acceleration shown by Pioneer is not observed in the

planets may be explained: The anomalous acceleration is not real, it is an artefact affecting

Doppler measurements of bodies which are in a place where the index of refraction n’ ≠ 1

and are in relative acceleration to Earth-bound observers. A Doppler probe on the surface

of the planets will show an anomalous acceleration because the energy density of space

there is different from the energy density on the surface of Earth. Hence the index of

refraction n

′

on the surface of planets differs from Earth. Table I shows the results of

calculating

n

′

with the use of Eq. (7). The values close to 1.0 being caused by the local

gravitational energy density being not so different from the surface of the Earth. Values of

n

′

above one indicate that a Doppler probe would show an anomalous acceleration in the

direction opposite to the Sun and would be equal to the factor

)'1( n

−

times the real relative

acceleration of the planet.

Table I. Values of the index of refraction

'n in the surface of the planets and the

Moon. The value of

*

ρ

= 1.09429 x 10

15

Joule/m

3

was used in evaluating 'n with

equation (7).

Planet Mercury Venus Earth Mars Jupiter

'n

0.99997382

0.99999527

1.00000000 0.99997758

1.00014145

Planet Saturn Uranus Neptune Pluto Moon

'n

1.00000349

0.99999524

1.00000737

0.99997385

0.99997454

11

8.- Conclusion

We find a neo-Newtonian explanation of the Pioneer anomaly. This is done with the Curé

[10, p. 173] hypothesis that the speed of light at a site depends on the local space energy

density predicted by Newton’s universal law of gravitation. With this hypothesis we have

been able to deduce in a simple manner the empirically observed phenomenon of the

Pioneer anomaly qualitatively and quantitatively. Additionally with the theory developed

we are able to calculate the energy density of space produced by the rest of the Universe in

the neighbourhood of the Sun. The value obtained (1.0838. x 10

15

Joule/m

3

) coincides very

closely with a value (1.09429 x 10

15

Joule/m

3

) deduced by J. C. Curé [10, p. 279] on the

basis of the empirical measurement of light bending by the Sun observed during solar

eclipses.

The anomalous acceleration does not exist. Pioneer 10 and 11 as well as Galileo and

Ulysses spacecraft are moving according to Newton’s universal law of gravitation or

according to Einstein’s General Theory of Relativity which coincide in this respect. The

anomaly is found to be due to the effect on the Doppler signals by the index of refraction of

space, which is to say the variation of the speed of light due to the energy density of space

predicted by the Curé hypothesis.

For further verification of the Curé hypothesis we suggest careful analysis of measurements

done on the Pioneer spacecraft in the early stages of the flight, from launch to about 20 AU.

Fortunately there are plans at JPL, motorized by S. G. Turyshev, to reanalyze all the data

taken of the Pioneer missions, which have now been preserved. [8]

NASA’s careful measurements and the Curé hypothesis that the speed of light at a site

depends on the local space energy density which explain it have profound implications for

physics and cosmology. A lot of other astronomical data needs to be examined in this

context. Its acceptance on the basis of the evidence supplied by an explanation of the

Pioneer anomaly and the light bending by the Sun obliges a careful revision of the

interpretation of data used by Hubble to derive the hypothesis of the expansion of the

universe and all the theoretical predictions which follow.

Acknowledgements

I would like to thank my colleagues Haydn Barros, Imre Mikoss and Guillermo Chasín for

helpful discussions during the development of this work and Gabriel Bernasconi for recent

literature on the Pioneer anomaly. Also thank Simon E. Greaves for independent

calculation of the numerical values and to Jorge C. Curé for pointing out that the Pioneer

anomaly could be explained on the basis of work in his book.

References

[1] Anderson, J. D., Laing, P. A., Lau, E. L., Liu, A. S., Nieto M. M., and Turyshev. S. G. Indication, from

Pioneer 10, 11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration.

Phys. Rev. Letters. 81(1998) 2858-2861[Comment by Katz J.I.: Phys. Rev. Lett. 83, 1892 (1999); Reply: Phys.

Rev. Lett. 83, 1893 (1999)].

[2] Turyshev, S. G., Anderson, J. D., Laing, P. A., Lau, E. L., Liu, A. S. and Nieto, M. M. The Apparent

Anomalous, Weak, Long-Range Acceleration of Pioneer 10 and 11. arXiv:gr-qc/9903024 v2 9 (Mar 1999).

12

[3] Nieto, M. M. and Anderson, J. D. Using Early Data to Illuminate the Pioneer Anomaly. LA-UR-05-5110

v2 9 Mar. 1999 (October 8, 2005).

[4] Anderson, J. D., Laing, P. A., Lau, E. L., Liu, A.S., Nieto, M.M. and Turyshev S.G. Study of the

anomalous acceleration of Pioneer 10 and 11. Phys. Rev. D 65, 082004 (2002). arXiv: gr-qc/0104064 v5

[5] Conference on The Pioneer Anomaly- Observations, Attempts at Explanation, Further Exploration -

ZARM, Bremen (18 - 19. May 2004).

http://www.zarm.uni-bremen.de/Pioneer/

[6] Pioneer Anomaly Conference in Switzerland. (November 6-10, 2005). The Planetary Society. Merek

Chertkow comments in: http://www.planetary.org/programs/projects/pioneer_anomaly/update_200511.html

[7] The tiny magnitude of the calculated acceleration is more easily understood expressing it in the familiar

length unit used for atoms and molecules: The Angstrom. The acceleration is about 8 Angstroms/s

2

.

[8] Clark, S. Fly another day. New Scientist. 3 June, 2006. p. 46-49.

[9]

Shiga, D. Fly-by may be key to Pioneer anomaly. New Scientist. 19 August 2006. p.13.

[10] Jorge Céspedes-Curé Einstein on Trial or Metaphysical Principles of Natural Philosophy. Publisher: et

al. Organization, 2002. Available at Amazon.com; Amazon.co.uk.

[11There are other wildly different estimations of the “Zero point energy density of vacuum” an analogous

concept within quantum field theory.

[12] By

ρ

* the interstellar primordial energy density in the vicinity of the Sun we mean within a few light

years of the Sun.

ρ

* is probably not a Universal constant but rather a spatial function of the star density in the

vicinity. Higher near galactic centers, lower in intergalactic space and decreasing with distance from the

center of the universe.

[13] Merat, Parvis. Analysis of the optical data on the deflection of light in the vicinity of the solar limb.

GRG. 5, No. 3, pp757-764 (1974)

[14] Examination of the primary papers on the Pioneer anomaly do not give the measured Doppler drift

frequency as a function of distance from the sun of the spacecraft. This information is undoubtedly contained

in JPL or NASA reports. What is reported is the interpretation of the Doppler drift frequency as an

unmodeled acceleration towards the Sun as a function of heliocentric distance. The data used in Fig. 2 was

derived from Fig. 7 of Ref. [4].