ARGUMENTS

John D. Aderson et al. [I]

- Conclude to the existence of a radial acceleration aP 
aP = (8.74 +- 1.25) x 10^-8  in order to explain  the observed
redshift deficit:

"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 supporting data from the Galileo and Ulysses spacecraft."

"The observed two-way anomalous effect may be expressed
by the following simple expression:
NUobs(t) = NUmodel(t) [1 - 2aP*t/c],        
where NUobs is the two-way Doppler frequency shift of the
re-transmitted signal observed by a DSN antennae, while
NUmodel is the predicted frequency shift of that signal."

- But the so-called radial acceleration is not necessarily sunward:

"We only measure Earth-spacecraft Doppler frequency and, as 
we will discuss in Sec. 8.1, the down link antenna yields a conical 
beam of width 3.6 degrees at half-maximum power.
Therefore, between Pioneer 10's past and present (May 2001) 
distances of 20 to 78 AU, the Earth-spacecraft line and Sun-spacecraft
line are so close that one can not resolve whether the force direction 
is towards the Sun or if the force direction is towards the Earth. 
If we could have used a longer arc fit that started earlier and hence
closer, we might have been able to separate the Sun direction from 
the Earth direction."

They also wrote: "The anomalous acceleration is too large to have 
gone undetected in planetary orbits, particularly for Earth and Mars."

In conclusion, the observed acceleration can be opposite the velocity 
rather than sunward.

- Don't find a convincing explanation for the observed acceleration,
and consider that "the possibility remains that the effect 
is real, and could even be related to cosmological quantities."

Indeed, it is in good accordance with cH, where c is the speed 
of light and H a Hubble constant of 82 km/s/Mpc.

- Don't exclude that the anomly could lead to a new physics:

" If one has to consider new physics one should be open to both
points of view. In the unlikely event that there is new physics, one
does not want to miss it because one had the wrong mind set."

PROPOSED BASE OF A NEW PHYSICS

- The observed acceleration is real.

- It is equal to cH, hence it is a cosmological phenomenon,
that affects all moving bodies. In other words, it is related to 
the Universe as a whole, not to the Sun or its environment  
in particular.

- Consequently, planets experience a tangential deceleration,
and light undergoes a red shift in proportion to the distance 
of its source.

-  Assuming that the acceleration is constant, the simplest
explanation is that the Universe is Euclidian, homogeneous, 
static and infinite, and that every point of the Universe is the 
center of a sphere of radius c/H. 

POSSIBLE OBSERVATIONS

The effect of a tangential deceleration cH on the orbit of
planets or satellites is very small.

The used formula is h-h' = (at)^2 / g(h), where h is the initial 
height of the satellite, h' its height after a period of time t, 
a the constant deceleration cH and g(h) the gravitational
potential at h. This formula is valid if if the difference h-h' 
is small.  [II]

For instance, the orbit of the Moon would be reduced to
3/4 of its present value after about 9 billion years !  
The change is definitely not linear, but 1cm/year is a good
approximation for an orbital decrease of about 100000 km.

According to LLR data, the Moon is receding from Earth at a
rate of about 3.8 centimeters per year. Such increase of
orbital distance, attributed to tidal effects on Earth, would mask
the present small decrease of 1.87 cm/year. So, the orbit of 
the Moon cannot be used to prove the existence of a cosmological 
deceleration cH.

But the non-Newtonian component of the precession of Mercury 
(about 43" / century) could be explained by such tangential
deceleration, because the decrease rate of its aphelion shoud be
significatively greater than that of its perihelion.

Marcel Luttgens
-----


[I] Excerpts from "Study of the anomalous acceleration of
Pioneer 10 and 11", by John D. Anderson, Philip A. Laing,
Eunice L. Lau, Anthony S. Liu, Michael Martin Nieto
and Slava G. Turyshev.
(arXiv: gr- qc/ 0104064 19 April 2001)
----------

It is also of interest to consider some of the recent proposals
to modify gravity, as alternatives to dark matter [125]-[128].
Consider Milgrom's proposed modication of gravity [128],
where the gravitational acceleration a of a massive body is
proportionnal to 1/r 2 for some constant a0<>a. Depending 
on the value of H, the Hubble constant,
a0=+-aP ! Indeed, as a number of people have noted,
aH = cH, or 8E-8 cm/s^2 if H=82 km/s/Mpc.

Of course, there are (fundamental and deep) theoretical
problems if one has a new force of the phenomenological types
of those above. Even so, the deep space data piques our curiosity.
However, these and other universal gravitational explanations
for the Pioneer effect come up against a hard experimental wall.

The anomalous acceleration is too large to have gone
undetected in planetary orbits, particularly for Earth and Mars.
NASA's Viking mission provided radio-ranging measurements
to an accuracy of about 12 m [129, 130]. If a planet experiences
a small, anomalous, radial acceleration, aA, its orbital radius r
is perturbed by delta(r) = - l^6*aA/(GM)^4 ---> - r*aA/aN (56), where
M is the solar mass, l the orbital angular momentum per unit
mass and aN is the Newtonian acceleration at r. (The right value
in Eq. (56) holds in the circular orbit limit.)

For Earth and Mars, delta(r) is about -21 and -76 km. However,
the Viking data determines the difference between the Mars and
Earth orbital radii to about a 100 m accuracy, and their sum to
an accuracy of about 150 m. The Pioneer effect is not seen.

Further, a perturbation in r produces a perturbation to the
orbital angular velocity of delta (omega) = 2l*aA/GM --->
2phi*aA/aN.

The determination of the synodic angular velocity (omega(E) -
omega(M) is accurate to 7 parts in 10E11, or to about 5 ms
accuracy in synodic period. The only parameter that could
possibly mask the spacecraft-determined aR is (GM).
But a large error here would cause inconsistencies with the
overall planetary ephemeris [9, 47]. [Also, there would be a
problem with the advance of the perihelion of Icarus [131].] 

We conclude that the Viking ranging data limit any unmodeled
radial acceleration acting on Earth and Mars to no more than
0.1E10-8 cm/s^2. Consequently, if the anomalous radial
acceleration acting on spinning spacecraft is gravitational in
origin, it is not universal. That is, it must affect bodies in the
1000 kg range more than bodies of planetary size by a factor
of 100 or more. This would be a strange violation of the Principle
of Equivalence [132]. (Similarly, the delata(omega) results
rule out the universality of the at time-acceleration model.
In the age of the universe,T, one would have a(t)T/2 ---> 0.7 T.)

[.....]

Finally, we observe that , as has
been intimated. (See Section 11, especially the text around
Eqs. (55) and (58).) This possibility necessitates a cautionary
note on phenomenology: At this point in time, with the limited
results available, there is a phenomenological equivalence
between the aP and at points of view. But somehow, the choice
one makes a affects one's outlook and direction of attack.
If one has to consider new physics one should be open to both
points of view. In the unlikely event that there is new physics, one
does not want to miss it because one had the wrong mind set.>>

They also wrote:
"It is important to realize that our experimental observable is
a Doppler frequency shift, i.e., NU(t).[See Figure 8 and Eq.(15).] "
and, at page 31:
"The observed two-way anomalous effect may be expressed
by the following simple expression:
NUobs(t) = NUmodel(t) [1 - 2aP*t/c],               (15)
where NUobs is the two-way Doppler frequency shift of the
re-transmitted signal observed by a DSN antennae, while
NUmodel is the predicted frequency shift of that signal."


From http://arxiv.org/PS_cache/gr-qc/pdf/0104/0104064.pdf:

The arguments for gas leaks" are:

i) All spacecraft experience a gas leakage at some level.
ii) There is enough gas available to cause the effect. 
iii) Gas leaks require no new physics. 
iv) However, it is unlikely that the two Pioneer spacecraft would have gas
leaks at similar rates, over the entire data interval, especially 
when the valves have been used for so many maneuvers. 
[Recall also that one of the Pioneer 11 thrusters became inoperative
soon after launch. (See Section 2.2.)]
v) Most importantly, it would require that these gas leaks be precisely
pointed towards the front [19] of the spacecraft so as not to cause a 
large spin-rate changes. 
But vi) it could still be true anyway.

The main arguments about "heat" are:

i) There is so much heat available that a small amount of the total 
>could cause the effect. 
ii) In deep space the spacecraft will be in approximate thermal
equilibrium. The heat should then be emitted at an approximately
constant rate, deviating from a constant only because of the slow
exponential decay of the Plutonium heat source. It is hard
to resist the notion that this heat somehow must be the origin of the
effect. However, 
iii) there is no solid explanation in hand as to how a specific heat
mechanism could work. Further, 
iv) the decrease in the heat supply over time should have been seen 
by now.

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 aP = +- 8E-8 cm/s^2 directed towards the Sun [1, 2].
Much effort has been expended looking for possible systematic 
origins of the residuals. We find that it is difficult to understand how 
any of these mechanisms can explain the magnitude of the observed
behavior of the Pioneer anomaly. 
[...]
There are now further data for the Pioneer 10 orbit determination.
The extended Pioneer10 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 aP = (8.74 +- 1.25) x 10^-8.

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 supporting data from the Galileo and Ulysses spacecraft.

[72] We only measure Earth-spacecraft Doppler frequency and, as 
we will discuss in Sec. 8.1, the down link antenna yields a conical 
beam of width 3.6 degrees at half-maximum power.
Therefore, between Pioneer 10's past and present (May 2001) distances of 20 to
78 AU, the Earth-spacecraft line and Sun-spacecraft
line are so close that one can not resolve whether the force direction 
is towards the Sun or if the force direction is towards the Earth. 
If we could have used a longer arc fit that started earlier and hence
closer, we might have been able to separate the Sun direction from 
the Earth direction.


[II] ORBIT CALCULATION

To calculate the effects of the deceleration on the orbit
of planets (or satellites), I have developed the following
Qbasic programme that gives the formula linking a constant 
tangential deceleration of a planet (or satellite) and the distance 
of the planet from Earth or Sun at times t1 and t2.
I am grateful to Gregg Neil and Brian Davis, who gave me
their views about such calculation.


Greg Neill:
 
"Let's say the satellite orbit remains near circular, and that
the deceleration is understood to be a constant loss of orbital
energy over time.

For any orbit we have:

  (V^2)/2 - u/r = E               (eq.1)
      where:
            V is the speed of the satellite
            r is the distance of the sattellite
              from the center of the Earth
            u is the gravitational parameter G*Mearth
            E is the total specific energy

For a circular orbit we also have:

  u/(r^2) = (V^2)/r      {Gravitational accel. = centrifugal accel.) (eq.2)

so that V^2 = u/r

Substituting for V^2 in eq.1 we have:

  u/(2r) - u/r = E

  r = -u/(2E)                     (eq.3)

There is a -ve on the right hand side of the above because the
specific energy for a bound orbit is negative, and two negatives
make a positive for the radius.

The deceleration represents a continual loss of energy of the satellite.
Write your deceleration in terms of energy loss per unit time and
figure the total energy lost to friction between your times of interest.
Then use eq.3 to find the corresponding orbital radii."


'ORBITAL DECREASE OF SATELLITES

' h-h'= [(R+h)(R+h') / GM] * (at)^2, or
' h-h' = 1 / [(g(h)/(at)^2) + 1/(R+h)], where g(h) = GM/(R+h)^2

'where R and M are the radius and mass of the Earth, G the 'gravitational
'constant, h the initial height of the satellite, h' its height
'after a period of time t, and a the constant deceleration cH.

'If the difference h-h' is small, the formula reduces to
'h-h' = (at)^2 / g(h)

'One has to calculate the orbital decrease "delta" per revolution
'to estimate the yearly decrease  "deltayear" ( by multiplying 
' "delta" by the number of revolutions per year "norb")

'Example: Earth/Moon

CLS 
DEFDBL A-Z

pi = 3.1416
M = 5.977E+27     'Earth mass in grams
G = 6.6732E-08

R = 6.378E+08     'Earth radius in cm
h = 3.84E+10        'mean distance Earth-Moon in cm  
a = 8E-08              'constant deceleration (cf. Pioneer)
t = 3.156E+07       'one year (in seconds)
tm = 86400!           'one month (in seconds)

d = R + h         'orbital radius
torb = 2 * pi * d / SQR(G * M / d)   'duration of one revolution in sec
norb = t / torb                                   'number of revolutions/year

gh = G * M / ((R + h) * (R + h))
delta = a * torb * a * torb / gh    'delta = h-h' in cm
deltayear = delta * norb             'delta year = orbital difference in cm
day = 3600 * 24!                        'duration of one day in seconds
tday = torb / day                         'duration of one revolution in days

PRINT "duration of 1 revolution in days"; tday
PRINT "delta="; delta
PRINT "deltayear="; deltayear

END

1) Some results for different distances Earth/Moon

h = 3.84E+10       delta = 1.873 cm / year
h = 1.92E+10                    0.18 cm
h = 0.96E+10                    0.017 cm
h = 0.48E+10                    0.00189 cm

According to LLR data, the Moon is receding from Earth at a 
rate of about 3.8 centimeters per year. Such increase of
orbital distance, attributed to tidal effects on Earth, could mask 
the present small decrease of 1.87 cm/year.

2) Effects of a tangential deceleration of 8E-8 cm/s^2 on the
    orbit of Mercury:

The aphelion of Mercury decreases at a rate of about 1m/century,
whereas its perihelion decreases by only 0.13 cm/century.
Such difference could explain the non-Newtonian component
of the precession of Mercury (about 43" / century).

Marcel Luttgens
June 16, 2001