On Dec 24, 6:51 pm, Eric Gisse wrote:
On Dec 24, 7:07 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.

...and your justification for this is what, exactly?

Remember what the Schwarzschild solution describes.

Modeling the Sun as a stationary point mass, and computing the time
dilation effects of the Sun within the Pioneers' paths ( 100 a.u.). A
point mass iss certainly accurate enough when we get as far away as
the earth, no?

Note that any mass will cause time dilation according to GR. With
black holes the time dilation is massive near the event horizon, but
common and garden bodies like the Sun (and even a tennis ball) will
cause similar (although much smaller) time dilation effects.

Roland