The Anomalous Acceleration of Pioneer 10 toward the sun of
about 10^-8cm/sec^2 at various distances r from the sun can be
associated with
the fact that the velocity of the spacecraft is greater than the
orbital velocity the spacecraft would have in a circular orbit at
the same distance.
This is necessary so that the spacecraft will not be made to
move
in orbit about the sun and could escape the solar system
but it also implies that the attractive mass of the
spacecraft is greater than it otherwise would be.
Is the increase in mass consistent with the increase
in mass with velocity according to Einstein:
m=(m_0)(1-v^2/c^)^-1/2?
No but there is a similarity if you think of the velocity
of the spacecraft at a point in space not relative
to the speed of light but to the orbital velocity
at that point in space.

How can an objects mass increase in this way?
It is possible to describe the gravitational
attraction between objects on the earth toward the center of the
earth
and of objects in the solar system toward the sun etc in terms of
electrostatic
dipoles in the objects radially oriented to these respective
centers.
The dipole dipole force varies inversely as the fourth power
of distance
while the gravitational force is an inverse square force. But if
the size of the
dipole is proportional to the distance of the dipole from another
dipole
eg a dipole in the spacecraft and a dipole in the sun then the
dipole dipole
force reduces to an inverse square force. The dipole in the craft
is due to
the net effect of dipoles in each element, each proton and
neutron and the
dipole in the sun is due to the net of the dipoles in each
element of the plasma
sun.
The plausibility of electrostatic dipoles interacting in this
way is shown as due to the decrease in interference between
the dipoles as distance increases. That is, the primary
determination
of the size of the dipole is its speed. Its speed is caused by a
force
which produced and may continue to produce an
accleration and at the same time an increase of
polarization of charge in its elements eg protons and neutrons
etc..
The details are given in http://www.bestweb.net/~sansbury.


The speed of the craft,now 12km/sec according to Pioneer
home page was about 36.67km/sec as it passed Jupiter while
29km per sec relative to the sun when it was on earth
orbiting the sun.
If the spacecraft was in orbit around the sun at a distance
r from the sun it would have an orbital velocity of v from
GM/r^2=v^2/r So its orbital velocity at a distance r can
be compared to its actual velocity v*_r compared to v_r.
The Pioneer 10 spacecraft is moving almost completely
radially away from the sun such that the sine of the angle
between its
trajectory and a radial line to the sun is very small eg .001.
The spacecraft is also free to rotate. According to this
hypothesis there would be a change in the attraction
of the spacecraft to the sun proportional to the difference
between (GM/r)^1/2 and v*_r. If r=10^12 then
((6.67)(10^-11)(1.99)(10^30)/(10^12))^1/2
=3.66(10^3.5)=11.57km/sec
about and the speed of the craft was probably more.
Hence the attraction

The attractive mass of an object on the earth directed to
the center of the earth is assumed to be due to electrostatic
dipole
inside protons and neutrons of length 10^-18 meters so that
(6.67)(10^-11) times [(1.67)(10^-27)]^2 = (9)(10^9)(es)^2 if
s=(.9)(10-18) is the gravitational force between two protons
one meter apart represented as the force between two
electrostatic
dipoles one meter part and colinearly and attractively oriented.
This gravitational force may in fact be due to the horizontal
component of the radial force between each proton and all
those on a radius from each toward the center of the earth etc.

And so the gravitational force between the sun and the earth
could be written as the force between radially oriented dipoles:
GmM/R^2 = 9(10^9)mM[6.02)(10^26)]^2 times kK times s*S*
times (2.56) times 10-38 divided by R^2 where the dipoles are es*
and
eS* and e=1.6(10^-19)Coul.;this implies kKs*S*=
(.0079)10^(-61-11+38) =
10^-36 approximatelySince the Sun is .75H+.25He so that 1.75kg
of Sun contains 6.02 times 10^26 molecules each of which contains
on average 1.75 protons+neutrons so 1kg of the gaseous Sun
contains 6.02 times 10^26 protons+neutrons in a volume that is
larger of course than that of 1 kg of a solid planet; but 1kg of
any planet or the Sun contains the same number of
protons+neutrons. There are about 2(10^30) kg in the Sun. Hence
the Sun contains 6.02 times 10^26 times M or 12 times 10^56 and
the Earth contains 6.02 times 10^26 times m or 3.59 times 10^51
unit dipoles in the Earth. The total dipoles a
1.2(10^57)k(s)RS* and 3.59(10^51)K(S)Rs*.
Hence . Now RkS* and RKs* are the magnitudes of the dipoles
associated with the Sun and planet respectively where R varies
from around 1.5(10^11)meters ( 10^10 to 10^13 meters for the
planets)
But we also know that the Earth's dipoles cannot be much larger
than atomic
nuclei about 10^-15meters =RKs* that Ks*=10^-26 which implies
kS*=10^-10 and also RkS*= 10^(-10+11) so the dipoles on the
Sun would have to be 10 meters in length or the amount of
charge in
each dipole is more than e=^-19 etc.
We assume, following the Wilson Bartlett relation between
angular momentum and gravity,
that dipoles in protons and neutrons on planets that produce
their attraction to the sun is
due to the orbital speed of the planets and so a part of the
planet, like the spacecraft, when moving apart from the planet
at a different speed
will have its dipoles change and so its attractive mass will
change. see http://www.bestweb.net/~sansbury