Fresh edited: 29-1-2012
(Data and calculations will be updated later, there are at-least 40 pages of hand written equations and calculations which saw me through this, of which 50% I had uploaded about a week ago, 25-11-2011)
I have “cracked” the spinning earth effect on flyby: Galileo-I, from constant of motion (angular momentum) and conservation of energy consideration (this is why I called Weinberg my attaboy, 6 yrs ago, his text is a mervel=reference Weinberg’s text). He wrote his text in 1970, 2 decades before flyby anomaly was observed and 2 decades later of this anomaly I am seeing an explanation from basic relativity theory which was achieved a century ago. Add a function , I have formulated everything in terms of , so all effects vanish at infinity. Note that this compares with of radial velocity of recession of the satelite at infinity which is 13.7 km/s. In that case the spining earth is at-least 13 times larger than the radial recession effect. At or near perigee, it will change, but will be perhaps larger. Now you see where my 1.3% effect of spinning earth was coming? Perhaps, I haven’t checked.
The explanation of flyby anomaly: our radars are placed on the surface of earth, which must measure the effects of the earth’s potential of on surface (nothing but ~ velocity squared) and the ~1.3% spinning effect which is always present on surface of earth. This translates to about 3.92 mm/s for Galileo-I which does not affect the radar when it is infinitely away. The special relativity effect must be zero at infinite because it is infinitely away. The other satelites are exhibiting different value because of telemetry frequencies and parameters specific to their motion, eg the the satellite mass and exact trajectory. If you change your radar-frequency the velocity will be registered proportionally. Note that earth gravitational potential (energy) entails a GHz frequency for “visibility” of the speed increament, in consistent units. In Relativity, we measure distance to millimeter accuracy and flyby-observation is a witness to that accuracy. So also, GPS satelites providing millimeter accuracy is not a shock. (reference to OPERA GPS accuracy)
If telemetry is done using 0.5 GHz to 1 GHz (at-least a similar variation) by the team who tracked the Galileo-I flyby, I am essentially done with solving the paradoxical anomaly. There is no anomaly, together 1. gravitational shift 2. earth spin 3. general Doppler shift, all angles not just transverse, explains it very well. I don’t have the exact function for earth spin yet but a put by hand percentage which one can vary and one does not see much change after the required value is obtained. Also it was an unresolved anomaly until now, for two decades. There was only an empirical fitting in terms of angular deviation (basic celestial mechanics actually, reference Weinberg’s text) by Anderson (2009), for all satelites and in another paper and article there was only a slight mention (no calculations) that only Doppler shift would suffice no gravitational shift is needed, (another paper, not Anderson’s, need to recover the reference).
That is not true, we need all the effects even the few % earth spin is needed to bring it very close and exact telemetry used, is necessary for exact fitting. We are seeing so much variation in the Galileo-I data, perhaps they kept on tuning their radars to catch the satelite, in a varying frequency and this refelects as: much variation in the velocity of the satelite from telemetric observation. They should have kept that data. It was the first ever flyby observed (? YES) so they were all excited, but science does not work on excitation but perseverance. I took few hours to construct a well-behaved telemetry frequency adjustment, one of them worked out well enough at short distances, but if it is not what was used it only biases the over all result. Also if data from Galileo-I is available only then one can study in detail the total implication with exact error analysis. It is such an important study because it tests Theory of Relativity which is at the base of every modern physics studies, research, technology, engineering and philosophy we do, that not having this data would be a shame.
Eureka: flyby anomalies are a great way to measure the earth spin time dilation. because speed increament at infinity is very flat the earth spin effect becomes closer to its true value, which in my analysis I find to be 1.3%, net spinning acceleration is ~1%, remember: centripetal and coriolis (reference another article of mine). This % depends on how you are measuring it, frequency shift, speed-shift or what frequency you are using in your telemetry. I theorized for a 1 GHz frequency, in terms of speed this turns out to be observed 3.92 mm/s from my calculated shifts if I adjust by 1.3% earth spin speed reduction. At infinity earth spin is present in opposite sign alternatively over infinitesimal changes, at close-up the alternation may occur alternatively at larger variation of distance. One more thing is I have used large inclinations of flyby trajectories whose cosines are in the denominator, that is, it amounts to more than 1 GHz, as these 1.25 and so on factors are present in the computation in almost all terms. When you send a frequency at higher range does it get dissipated when it travels long distances and come back? Then you are going to see the effect at a lowered frequency and so on. So a frequency theorized approach is more ideal than how we translate it back to speed increament. (as I have formulated it) We see fluctuation in terms of what exact frequency is registered in the observation.
SO at infinity one must subtract earth-spin effect for both prograde and retrograde motion of satelite? (yes). For ferquency shift or time-dilation one obtains twice this, that is earth spin is 2.6% which makes sense since earth angular speed is large at large separations where the effect of gravity is vanishingly small because of Newtonian and Minkowskian 1/r potential. What difference does it make to a distant star that the earth is spinning. by the time the star-light reaches here the earth has made a great many number of rotations but the distance is so large the static effect itself has vanished. spin all you can but you do not delay my clocks any more because I am not bound to your gravity any more, says Galileo. Galileo recently broke up with earth and now happily lives with Jupiter along with it’s other wives IO, Europa, Ganemede and ? There are no churchs upthere either to torment Galileo for heresy. Galileo left earth when born and fell into the lap of Jupiter after it deceased.