On Dec 24, 8:07 am, Roland PJ wrote:
The Pioneer Spacecraft have shown an anomalous acceleration towards the
Sun of ~8 x 10^-16 m/s^2.

Pioneer 10 and Pioneer 11 were traveling at about 1.22 x 10^4 m/s and
1.16 x 10^4 m/s away from the Sun at the end of their lifetimes.

Now, the Pioneers will have experienced time dilation in the correct
direction (blue-shift for us on earth) as they escape the gravitational
well of the Sun, as follows:

We will use Swarzschild time dilation, dt_shell = (1-2M/r)^1/2 dt, where
dt_shell is shell time (i.e. use on earth, or Pioneer's electronics),
and dt is ephemeris time (bookkeeper time) valid at a great distance.

The mass of the Sun is 1.989 x 10^30 kg, or 1.477 x 10^3 m (in
length-units, valid for the above equation). I'll use 1.5 x 10^3 m in
the following.

And the earth is 1 a.u., or 1.496 x 10^11 m, from the Sun. I'll use 1.5
x 10^11 m in the following.

So, time dilation on the earth relative to ephemeris time, is:

dt_e ~= (1 - 2.1.5 x 10^3 / 1.5 x 10^-11) ^ 1/2 dt
= (1 - 2.10^-8)^1/2 dt
~= (1 - 10^-8) dt

And time dilation on Pioneer, when at n a.u. from the Sun (i.e. n times
the earth's distance from the Sun) is:

dt_p ~= (1 - 2.1.5 x 10^3 / n.1.5 x 10^-11) ^1/2 dt
~= (1 - n.10^-8) dt

So, time will 'speed up' on Pioneer relative to the earth at a rate:

dt_e/dt_p = (1 - 10^-8) / (1 - n.10^-8), where n is in a.u.
(astronomical units).

So, the rate of change of relative time dilation between Pioneer and
earth, per a.u. is ~10^-8 / a.u. (differentiate and approximate), as
long as Pioneer is not too far away (certainly less than 10^4 a.u.).

Now, Pioneers move at about 1.2 x 10^4 m/s, which is about 1.2 x 10^4 /
1.5 x 10^11 a.u./s, or ~ 8 x 10^-8 a.u./s.

So, the apparent rate of relative time dilation of the Pioneers, as seen
by earth will be about:

10^-8 x 8 x 10^-8 /s

= 8 x 10^-16 / s.

This corresponds to an apparent acceleration of 8 x 10^-16 m/s/s of the
Pioneer craft as viewed from earth, which is in excellent agreement with
the measurements of just over 8 x 10^16 m/s/s, normally presented as ~8
x 10^-13 km/s/s.

Actually I think the last step of my logic is deeply dodgy, but the
nunmbers were in such good agreement that I felt compelled to ignore my
reservations

LOL, "compelled". the thing about the anomally
is it's ratio (acceleration), somewhere around
1/1700. Usually SR and GR effects are proportional
to v^2/c^2 (or 2 phi/c^2), this effect is more
like 3Vescape/c or something like that,(my papers
are filed), that's actually a pretty large
effect to be handled by "known" relativity.
Anyway, do your theory calculation and present a
formula. Then at the end plug in your values.
Regards
Ken S. Tucker

Roland