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GPS satelites static earth effect and spinning earth effect

November 22, 2011

Mohan, mdashf


View of CERN and Gran-sasso baseline from an altitude of 20200+9.08 kms, which is what the altitude of GPS satelite is

earth at an altitude of 20200+9.08 kms. The red line I drew is a straight line of 696 kms 3-D straight line (perhaps non-Eucledian) near the CERN – gran sasso baseline parallel to the latter which is not straight and about 733 kms.

(isn’t that a reason to worry?)

This altitude is where the GPS satellites are “falling towards earth”. Why should different baseline ends not introduce any error to synchronization? Also if you imagine 13.7 kms/s at this altitude the satellite is almost not moving. It takes 2 minutes for a deviation of 1 degree angle on earth surface (or anywhere) which is “49000 time of flights” of a neutrino bunch from CERN to Grans Sasso. That is 3.54375 E-7 radians/time-of-flight. earth spin is 3.000823 E-2 radians/time-of-flight. (earth spin 7.292E-5 /sec). I showed that static earth effect is less than 1 pico-sec (special relativity effect is small and negative compared to general relativity effect). Spinning earth effect being 1% order of static earth, and earth spinning speed being E^5 orders higher than satellite angular speed again the “1% general relativistic effect of 1 pico-sec” will preside over the special relativistic effect of the satellite. SO the non-inertialness of earth will be ~1% of static earth which is 1-picosecond. In other words the total relativistic effect of the GPS satellite considering even earth spin will be within about 1-pico-sec. We need not worry about this.

HERE is a diagram that will make it more concise: ((some calculational mistakes, does not change the idea, will fix later, check next diagram with correct calculation, error in earth-spin and satelite spin/tof, but satelite spin is already present in static earth calculation, that is the 8 meters/tof is already produced an effect less than 1 pico-second which was static earth gravitational potential on frequency emitted by satelite))

A completely new thinking: on earth any point can be considered to suffer from general and special relativistic effects and I have shown for GPS satelites, from their weak-field approximation, the static earth for any point has the highest effect about 1 pico-secs, the special relativity effect ... (READ below diagram)

the corrected earth and satelite spin at any point on surface, equitorial projections and trajectory around equator. The distance 733 kms is not equitorial distance. The 60 cms earth spin correction that OPERA does is because of this reason, there is angles whose cosine or sines are less than 1 for 733 kms distance. IF you can determine the unknown longitude x you can adjust the additional distance we have in our calculation and subtract it from 733 kms or whatever we have used all along, consistently. Then we will see a more correct result whethere OPERA is doing anything wrong or not. Hint: this triangle is a right angle traingle on the surface of earth, by construction.

(what a cool site I got in this respect, you can plug in my latitude, longitude, and you can get x and distances, what a cool .. 😉 http://www.movable-type.co.uk/scripts/latlong.html

NOTE: coordinates in lat and long shown above are from google earth for locations which are giving distance within few kms, so I need to plug in actual quoted lat and long from OPERA paper, later.

A completely new thinking: on earth any point can be considered to suffer from general and special relativistic effects and I have shown for GPS satelites, from their weak-field approximation, the static earth for any point has the highest effect about 1 pico-secs, the special relativity effect which comes from beta = 10^-5 order therefore negtive and less than the static effect. Thus total effect here is ~1 pico-sec. Then comes earth spinning which from principle of equivalence has a 1% acceleration effect hence equivalent static earth effect is 0.01 pico-sec. The special relativity effect here will again be less than this and negative to the general relativity effect because at any point on earth the beta of satelite motion is 10^-6 order to beta of earth-spin. The altitude of satelite does not matter as long as the signal from it reaches earth continuously, which is of nano-second range at-least, at any point, time of flight being 2.43 mili-sec. Note that the nano-sec is not playing any special role here, just that if we start with such precision and not do anyting silly we would not blow it up.

(The flyby satellites are hyperbolic and their effects vanish at very large distances from perigee, what remains is just static and spinning earth effects at earth surface where radars are placed)

SUGGESTION FOR OPERA:

One thing OPERA might have to do is: divide their 3 years data in a “Julian time arrow”, and place the actual time-coordinate during which neutrinos were released from CERN or registered at Gran-sasso, wrt all the satelite position-coordinate that were synchronizing the procedure on earth. This n-tuple if available for the entire data will be helpful in seeing if there is any deviation from time-synchronization as expected by a nano-second consideration that they claim. I assume such a n-tuple would be available in the total data they have.
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