Ranging and Pioneer

"Spud" wrote in message
oups.com...

gr-qc/0102103 is worth a look

I believe that analysis may be incomplete. It really
looks at the effect on ranging when using the round
trip time rather than the Doppler technique which
was actually used. The diagram at the end of the
paper shows the Earth moving round the Sun. I have
drawn a more detailed version showing the Earth and
the points of transmission and reception:

http://www.georgedishman.f2s.com/Pio...neerHubble.png

The two paths, red and blue, show individual wave
crests of the carrier [1] with the red path being
emitted first and with times subscripted with 'a'.
The times of events follow the convention used by
Anderson et al:

t1 transmit
t2 transpond
t3 receive.

The frequency of transmit signal is therefore

f_tx = 1 / (t_b1 - t_a1)

which is known of course, while that of the
received signal is

f_rx = 1 / (t_b3 - t_a3)

The speed (more precisely "range rate", the time
derivative of the path length) is inferred from
the ratio f_tx/f_rx [2] by

v = (f_tx / f_rx - 1) * c

hence

v = ((t_b3-t_a3)/(t_b1-t_a1)-1) * c

The anomalous acceleration is then the amount by
which the derivative dv/dt differs from the value
it would have in a non-expanding scenario where
H_0 = 0.

What I believe has been overlooked in gr-qc/0102103
is there will be an "expansion of space", notably
of the distance between the Earth and the craft in
accordance with the usual cosmological scale factor
a(t) during the time the signals are in flight.

I'm not up to handling GR but taking a macroscopic
view, I think the end result of the above should be
close to 2*H*v which is much larger than the range
value given in the above paper but still about four
orders smaller than the observed anomaly. It would
be interesting to know if that 'educated guess'
works out.

George

[1] This ignores the transponder ratio of 240/221
but is equivalent if the red and blue paths on
the uplink are cycles 221 apart while on the
downlink they are 240 cycles apart:

f_tx = 221 / (t_b1 - t_a1)

f_rx = 240 / (t_b3 - t_a3)

v = ( (240/221)*(f_tx/f_rx) - 1) * c

hence as above

v = ((t_b3-t_a3)/(t_b1-t_a1)-1) * c

[2] This is the opposite of the normal convention
as speeds away from Earth are treated as positive
by JPL but cause a reduction in received frequency.