This article should be read in conjunction with another which I just updated and reviewed extensively today (originally written 28-11-2011, reviewed 17-11-2013) but written about same time and due to same reason as this one written on (27-11-2011).
Flyby Anomaly and Earth’s Schwarzschild Radii are both millimeter order. It’s because they both come from Earth’s Gravitational Potential (which is of the order 10^-9 one should remember for all of life, one should be careful of units, eg G and c should be 1 or not). SO Flyby should never have been considered an anomaly in the first place. Call it a puzzle. Earth’s S-radii is 9 mm and anomalous speed ranges up-to 13 mm. Reasonable.
My detailed analysis gives exactly why S-radii determines this anomaly, S-radii = 2.m.G, at surface, speed anomaly is of order S-radii/actual-Radii. The analysis is extensive and most of it is on this web-journal along with most calculations. (search “flyby” in the tab on top and read 5-6 articles, in one most of the calculations are uploaded, when I have enough opportunities, I will update back-links if possible)
This will be written into a paper and all that has been studied will be organized when that task is taken up. This article can serve as a summary of central ideas. The more detailed calculations show “exact” value of the anomaly. A presentation of the Flyby-Anomaly is in the offing next week. (23-24 Nov 2013)
Note that all it means is since we are using light-speed in our theory we must make an error at the order of S-radii since by definition S-radii is the distance over which the relevant-mass does not let even light escape (read the linked article above and here: All masses behave like black-holes to some extent). SO thats (S-radii) the minimum error or speed anomaly. That is where the singularity of Theory of Relativity exists. Sun’s S-radii is 3 kms hence 3 kms/sec order error on satellites is permitted by Theory of Relativity, at the surface or away from surface the anomaly gradually drops down, as we go far from the effect of gravity, asymptotically to zero. Which is what we are seeing, the speed-anomaly of pioneer is decreasing. Except we do not go close to Sun’s surface hence we do not see 3 kms/sec. We are far away from it and seeing 1.27 mm/sec. (That is our radar is not placed on Sun’s surface but earth’s surface)
400 kms/year is not small, this millimeter accuracy is inbuilt into theory of Relativity, or it would not explain cosmology.
Schwarzschild radius (S-radii) is like the edge of a hole, if Gravity makes a hole in space time it would make an edge of the order of S-radii. So earth makes 9 mm edge in a space-time hole which gradually drops down as we go farther from earth. Sun makes an edge of 3 km and white dwarfs make even bigger edges. Call it the choco-pie edge of Universe. A proton makes an edge of ~10^-30 millimeter.
If a proton will shine off a photon (or even an electron) you would not know the trajectory of the light or electron by that amount, 10^-30 mm. But when the light or electron has escaped to far away distance the deflection would be much larger.
Needs review here
eg in 10^30 years the light would have deflected by 365*3600 milimeters = 1.3 million milimeters = 1.3 kms.
A proton lives for ~10^30 years. So in its life time it has misled zillions of photons.
What is the surface potential of a neutrino?
That would be ~2 x 10^-48. which is 10^-9 smaller than even the protons. So a photon can be misled by a neutrino,
Needs review here
~1.3 micrometers in 10^30 years.
How to prove theory of Relativity wrong for neutrinos?
Needs review here
Earth would deflect photons by ~9 mm in one second, 0.22 micrometers in 1 OPERA tof=2.43 milisecs and 540 pico-meters in 60 nanosecs.
Neutrinos will be deflected more if theory of relativity is correct for neutrinos, which are much more massive than light hence suffer more eccentricity. SO to prove theory of relativity incorrect you need nano-meter accuracy in nanosecond time accuracy. GPS provides millimeter accuracy. It registers the excess, does not prove the theory to be wrong.
I have stroked out any calculations that I am not comfortable at the moment and think they might have an error. The deflections need to be calculated as the light is moving at speed-of-light. So basic trigonometry will be applied to find this. I have also applied General Relativity equations to calculate the deflection. But it was used for a mass-less particle. For mass-have particles such as Flyby satellite Galileo-I I have calculated the details of speed-anomaly at infinite distance and at Perigee. But for deflection one needs to put in correct equations. This deflection is known [ref Weinberg’s text on gravity] and used by Anderson et al for an empirical formulation of all flyby anomalies without realizing why, which I had pointed out in one of the article on flyby anomaly. The incorrectness of deflection of satellite because of mass-less particle deflection in a gravity field is posted on this website. Perhaps back then I had pointed out that it was somewhat incorrect. But that will be review later as well.