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

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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/m3. A value which almost miraculously coincides with the same quantity: 1.09429 x 10^15 Joule/m3 derived by J. C. Cure from a completely different phenomenon: the bending of starlight during solar eclipses.
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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].
Article
Full-text available
For over 20 years NASA has struggled to find an explanation for the Pioneer anomaly, an unmodelled weak acceleration towards the sun (&ap; 8.5×10^{-10} m s^{-2}), observed in deep space probes Pioneer 10, 11, Galileo and Ulysses (Anderson et al. 1998, 1999; Katz 1999). No consensus explanation has been given since the anomaly was first announced, suggesting that new physics is involved. The riddle may be solved if we assume that c, the speed of light, is not a universal constant. Newtonian mechanics, together with the hypothesis by Céspedes-Curé (2002) that the index of refraction is a function of the gravitational energy density of space, leads to values of c slightly higher for interstellar space dominated by the primordial energy density rho_{*}^{} due to galaxies and far away stars, far from the gravitational influence of Earth, Moon, and Sun. The value derived for the index of refraction of space (n' < 1), implies a Doppler shift of the radio signal received from the probes which results in a decrease of the frequency received at Earth and interpreted as a weak acceleration towards the Sun.
Article
Full-text available
Recently we reported that radio Doppler data generated by NASA's Deep Space Network (DSN) from the Pioneer 10 and 11 spacecraft indicate an apparent anomalous, constant, spacecraft acceleration with a magnitude $\sim 8.5\times 10^{-8}$ cm s$^{-2}$, directed towards the Sun (gr-qc/9808081). Analysis of similar Doppler and ranging data from the Galileo and Ulysses spacecraft yielded ambiguous results for the anomalous acceleration, but it was useful in that it ruled out the possibility of a systematic error in the DSN Doppler system that could easily have been mistaken as a spacecraft acceleration. Here we present some new results, including a critique suggestions that the anomalous acceleration could be caused by collimated thermal emission. Based partially on a further data for the Pioneer 10 orbit determination, the data now spans January 1987 to July 1998, our best estimate of the average Pioneer 10 acceleration directed towards the Sun is $\sim 7.5 \times 10^{-8}$ cm s$^{-2}$. Comment: Latex, 7 pages and 2 figures. Invited talk at the XXXIV-th Rencontres de Moriond Meeting on Gravitational Waves and Experimental Gravity. Les Arcs, Savoi, France (January 23-30,1999). Corrected typos
Article
Full-text available
Our previous analyses of radio Doppler and ranging data from distant spacecraft in the solar system indicated that an apparent anomalous acceleration is acting on Pioneer 10 and 11, with a magnitude a_P ~ 8 x 10^{-8} cm/s^2, directed towards the Sun (anderson,moriond). Much effort has been expended looking for possible systematic origins of the residuals, but none has been found. A detailed investigation of effects both external to and internal to the spacecraft, as well as those due to modeling and computational techniques, is provided. We also discuss the methods, theoretical models, and experimental techniques used to detect and study small forces acting on interplanetary spacecraft. These include the methods of radio Doppler data collection, data editing, and data reduction. There is now further data for the Pioneer 10 orbit determination. The extended Pioneer 10 data set spans 3 January 1987 to 22 July 1998. [For Pioneer 11 the shorter span goes from 5 January 1987 to the time of loss of coherent data on 1 October 1990.] With these data sets and more detailed studies of all the systematics, we now give a result, of a_P = (8.74 +/- 1.33) x 10^{-8} cm/s^2. (Annual/diurnal variations on top of a_P, that leave a_P unchanged, are also reported and discussed.) Comment: Version 4, of 11 Apr. 2002, with errata appended
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Analysis of most observations of the deflection of light near the solar limb establishes the validity of general relativity's prediction beyond a certain distance of the sun and the existence of a new effect which is significant in the close vicinity of the solar limb.
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We illustrate the energy transfer during planetary flybys as a function of time using a number of flight mission examples. The energy transfer process is rather more complicated than a monotonic increase (or decrease) of energy with time. It exhibits temporary maxima and minima with time which then partially moderate before the asymptotic condition is obtained. The energy transfer to angular momentum is exhibited by an approximate Jacobi constant for the system.We demonstrate this with flybys that have shown unexplained behaviors: (i) the possible onset of the “Pioneer anomaly” with the gravity assist of Pioneer 11 by Saturn to hyperbolic orbit (as well as the Pioneer 10 hyperbolic gravity assist by Jupiter) and (ii) the Earth flyby anomalies of small increases in energy in the geocentric system (Galileo-I, NEAR, and Rosetta, in addition discussing the Cassini and Messenger flybys). Perhaps some small, as yet unrecognized effect in the energy-transfer process can shed light on these anomalies.
Article
Radio metric data from the Pioneer 10/11, Galileo, and Ulysses spacecraft indicate an apparent anomalous, constant, acceleration acting on the spacecraft with a magnitude $\sim 8.5\times 10^{-8}$ cm/s$^2$, directed towards the Sun. Two independent codes and physical strategies have been used to analyze the data. A number of potential causes have been ruled out. We discuss future kinematic tests and possible origins of the signal. Comment: Revtex, 4 pages and 1 figure. Minor changes for publication
Article
Analysis of the radio tracking data from the Pioneer 10/11 spacecraft at distances between about 20 - 70 AU from the Sun has consistently indicated the presence of an unmodeled, small, constant, Doppler blue shift drift of order 6 \times 10^{-9} Hz/s. After accounting for systematics, this drift can be interpreted as a constant acceleration of a_P= (8.74 \pm 1.33) \times 10^{-8} cm/s^2 directed towards the Sun, or perhaps as a time acceleration of a_t = (2.92 \pm 0.44)\times 10^{-18} s/s^2. Although it is suspected that there is a systematic origin to this anomaly, none has been unambiguously demonstrated. We review the current status of the anomaly, and then point out how the analysis of early data, which was never analyzed in detail, could allow a more clear understanding of the origin of the anomaly, be it a systematic or a manifestation of unsuspected physics. Comment: 19 pages, 6 figures, 2 tables, additional material
  • Phys
  • Rev
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)].
The Apparent Anomalous, Weak, Long-Range Acceleration of Pioneer
  • S G Turyshev
  • J D Anderson
  • P A Laing
  • E L Lau
  • A S Liu
  • M M Nieto
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).