The path of the Pioneer 10 spacecraft is not reproduced completely
accurately by Newtonian mechanics[1]; there is an excess
acceleration toward the Sun. Members of galaxies orbit consistently
faster than expected from the estimated baryonic matter inside their
paths[2]. Milgrom has modelled these deviations
empirically under the name MOND[3].
Hubble discovered that spectra of distant galaxies are redshifted
proportional to the independently estimated distance to the galaxies.
A plausible model of this effect is that the universe is expanding
uniformly. Freedman et al.[4] recently estimated
the expansion rate (Hubble constant) H as 72±8 km per second
per megaparsec, or 2.33±0.26×10^{18} sec^{1}.
Newton modelled orbital motions in the Solar System through a gravitational
force between objects of
F=G
m_{1}m_{2}
r^{2}
where F= the force of gravity, m_{i}= the objects'
masses, r= the distance between them, and G=6.673×10^{11} m^{3}/kg s^{2}.
For computing the trajectory of spacecraft through the Solar System,
experience shows that this model is extremely accurate. The Pioneer
10 spacecraft, at a distance from the Sun of about 67 astronomical
units (AU) or 1.0×10^{13} m, experiences a measured acceleration
toward the Sun of 1.32×10^{6} m/s^{2},
which is 8.74±1.33×10^{10} m/s^{2} less
than the Newtonian model[1, 5].
Galaxies are gravitationally bound, and their outer members revolve
considerably faster than might be expected from the detectable matter
in the galaxies. Milgrom[3] has also pointed out that
a similar adjustment to gravity, oriented to the center of the individual
galaxy, can explain their speed.
Because the universe is expanding, the spacecraft has an unmodelled
velocity of 2.33×10^{5} m/s. It takes
light and gravitation 3.34×10^{4} sec (0.4 day) to travel
from the Sun to the spacecraft. In that time the spacecraft moves
away by 0.78 m. In general, a distance of r is inflated by dr=r^{2}H/c,
where c=speed of light. To be precise about reference frames, we
measure (half) the roundtrip lightspeed time to the spacecraft, and
report a history of its postion 0.4 days in the past, which will be
r+dr where r refers to the distance when the telemetry
started out (and neglecting or subtracting off Newtonian modelled
motion)[6]. However, gravity propagates under the same
rules as light, and the spacecraft will experience an acceleration
as if its distance from the Sun were r. In other words, gravity
is the gradient of a potential computed thus:
φ(y
,t)=
∫
ρ(x,t

yx

/c)
(yx)^{2}
dx
where y is the position of the spacecraft, ρ(x,t)=the
density of matter as a function of location and time, and
integration is potentially over the whole universe. The density of
matter is recognized as it was before the lightspeed delay, when it
was closer to the spacecraft than the measured r+dr. As
a result, the Newtonian modelled acceleration will be too small by
2dr/r, which for Pioneer 10 comes out to 2.05×10^{19} m/s^{2}.
This correction is far too small to explain the anomalous acceleration.
Anderson et al.[1] point out that the anomalous
acceleration could be produced if the tracking receiver's master oscillator
slowed down relative to the transmitter's oscillator per the Hubble
constant. Specifically, the tracking system reports an acceleration
of a=hf(c/f) where h is the fractional frequency drift rate
in inverse seconds, f is the telemetry frequency in Hz (2.21×10^{9}
or 2.40×10^{9}Hz for the up and down links), and c is the
speed of light in m/s. Solving for h and using the downlink
frequency, we find h=2.92±0.44×10^{18} s^{1}
which is plausibly close to the currently measured value of Hubble's
constant. This result is independent of the distance to the spacecraft.
However, the uplink and downlink oscillators are the same hydrogen
maser, stable to 10^{15}f when integrated over 1000 seconds. If
the master oscillators at different tracking stations drifted at the
Hubble constant rate, it would have been noticed within hours. On
board the spacecraft, the uplink frequency is multiplied by a rational
number 240/221 to give the downlink, but if the downlink oscillator
unlocked and ran free, drifting at a rate h, the effect could be
explained, but a much larger and randomly changing drift would be
expected, which would be painfully obvious. Thus it is not credible
that actual frequency shifts in the equipment could have occurred.
Within experimental error the anomalous acceleration is constant between
radial coordinates of 15 to 45 AU (later extended to 67 AU), yet is
not significantly different from 0 in the range of 5 to 10 AU. A variation
of 20% was looked for (in connection with a possible source in the
Radioactive Power Generators, whose output decreases with time) but
was not found. Pioneer 10 and 11 show effects of the same size directed
toward the Sun (or Earth), though they travel in nearly opposite directions.
Thus, it is not credible that the acceleration, or appearance of acceleration,
is caused by an effect happening as the telemetry propagates through
space, which would be proportional to distance.
As for unmodelled masses in the Solar System, a constant acceleration
could be produced by a spherically symmetric cloud of matter whose
density varies as 1/r. Inside 67 AU its mass would have to be 1.3×10^{27}
kg or about 200 Earth masses. This is not credible.
Although the tiein with Hubble's constant attracts attention, a credible
connection cannot be discovered. Anderson et al.[1]
believe that the most likely cause of the acceleration is either unmodelled
gas escape, or the momentum of thermal photons from the Radioactive
Power Generators, differing from the model used.
John D. Anderson, Philip A. Laing, Eunice L. Lau, Anthony S. Liu,
Michael Martin Nieto, Slava G. Turyshev, ``Study of the anomalous
acceleration of Pioneer 10 and 11'', Phys.Rev. D65 (2002) 082004.
W. L. Freedman, B. F. Madore, B. K. Gibson, L. Ferrarese, D. D. Kelson,
S. Sakai, J. R. Mould, R. C. Kennicutt, Jr., H. C. Ford, J. A. Graham,
J. P. Huchra, S. M. G. Hughes, G. D. Illingworth, L. M. Macri, P.
B. Stetson, ``Final Results from the Hubble Space Telescope Key
Project to Measure the Hubble Constant", Astrophys.J. 553
(2001) 4772.
The acceleration is toward the center of mass of the Solar System;
transverse acceleration is 0±2×10^{10} m/s^{2}.
Within experimental error it is the same from a radial position of
15 to 67 AU, but from 5 to 10 AU it is different, not significantly
different from 0. The locations of the planets, Luna and larger asteroids
are modelled in detail. Minor gravitational effects are accounted
for generically. Numerous effects are accounted for in the position
of the telemetry antenna (such as Earth's rotational irregularities,
tides in the solid Earth, and continental drift), nongravitational
forces on the spacecraft (such as Solar radiation pressure, radiation
pressure from the telemetry transmission, and thruster leakage), and
conditions on the propagation path (such as interplanetary plasma
and atmospheric refraction). The quoted error of 1.33×10^{10} m/s^{2}
is the combined uncertainty in all these effects.
As for arc length, the outbound path is r+dr/2 whereas
the inbound path is r+3dr/2, and while the resulting phase
skew could potentially be measured, the position coordinate of the
spacecraft does not go by the arc length needed to reach it.