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Effect of Sun’s gravity on time dilation of atomic clocks on GPS satelite and on earth

November 3, 2011

Mohan, mdashf

Other planets can exert similarly understood effect, there will be no symmetry but for a 24 hrs period this may not matter. Their masses will be far smaller hence their effects as well considering their distances also play a role. WE can take only the massive and closer planets or make a table of such planets. This will be more complicated over the year but remember the satelite is moving with the earth so fractional effects and absolute effects do change but they belong to the time and configuration not to the essence of the calculation. That is a few more parameters can again be studied such as different solar dostances etc. Since the effects are supposed to be much smaller one can start with a simple classical calculation for distance and mass as known over the year.

actually you may also have to add earth’s radii of ~6000 kms at the beginning of the calculation below which will go both ways above and below Sun-earth distance without this offset. you need to from a rigorous satelite trajectory just integrate the potential energy over a 12 hrs multiply by 2 and see how it compares with the effect from sun on earth .

In the calculation below: there is an extra zero on sun-distance (figured out by dividing by speed of light, one must always get 8 mins)

I was thinking about the effect of the gravitational field of Sun on time dilation for atomic clocks present in it’s field. The atomic clocks that are placed in the GPS satelites make an orbit around earth every 12 hrs. That means half of the time the clocks on the satelite orbit on top of earth wrt Sun’s distance and the other half below it. But this difference is 52000 kms although Sun’s distance is 144000×10^3 kms from earth center to which you have to add Sun’s radii of 695×10^3 kms. SO you have the clocks from Sun’s center either (144695×10^3 – 26 x10^3) kms = 144669 x 10^3 kms or 144721 x 10^ 3 kms on top and below of earth. in second significant digit this is 1.45×10^8 kms, both ways, so it should not add any significant effect to the satelite clocks even over that distance, The effect that Sun’s mag field adds to times of clocks is DIRECTLY  proportional to the Sun’s potential energy at these distances and gravity potential energy of Sun is inversely proportional to distance. SO the effect is really really small since it will be of order 6.94 x 10^-11 (one needs to see what will be speed of light, c=1 or in m/s), OR you can see this in terms of inverse of all three distances which will give you 6.9123 x 10^-11 to 6.9098 x 10^-11. SO you can see the difference is really really small. (0.0025 x 10^-11, alarm?) Now compare with earth distance from Sun of 1.44×10^11 (center-center), effect is 6.9444×10^-11 (effect on clocks on earth, for which you can adjust another calculation for baseline distance of 733 kms, but this will be really insignificant compared to 26000 kms, what one has to do is aalso introduce the 6000 kms earth radii and subtract and add from Solar center and then introduce the 733 kms which will be altered every 24 hrs instead of satelite 12 hrs) I suggest we do the detail calculations since 6.9444 – 6.9123 = 0.0321 or 6.9444 – 6.9098 = 0.0346. I have greatly simplified this now, the only thing that remains is college calculations. Symmetry of situation is preserved in case of satelite in 12 hrs and incase of earth in 24 hrs, so we can calculate per the respective period and later multiply if needed a 2 for the satelite clocks. For other planetary bodies exerting gravity-potential energy to effect a time dilation as per general theory of relativity we need to also see that these planetary bodies are far less from solar mass.
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