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GPS time measurement is not affected by Earth’s Orbital motion around sun.

This is an original research work of Manmohan Dash, published here, 30 November 2011, 1.20 pm.

Why orbital motion of earth around sun, does not matter to GPS?

GPS-IIRM

Because, it is not only earth, the satellite is also marching in a circle around the Sun and the Sun is simply too far.

Sun and earth distance is 1 AU ~ 149,597,871 kms — 15 karod kms, mere pyare bhaion aur behenon.

Earth-Satellite distance is about 26,000 kms = 0.000174 AU ~2 milli AU.

The angular speed of earth and satellite, due to Sun, will be almost same; w = r^(-1.5) = sqrt (1/ r.r.r), what difference it makes, if we change r, by 0.0002, when r is 1, the result is 1 changes to 0.9997.

So, if the angular speed of earth, is 1, the angular speed of satellite, assuming it is 26000 kms closer to sun-center, would be 0.9997. Thats a difference in the 4th digit. Now angular speed of earth itself is of 10^-5 order. So, this is 10^-9 order change. Why do we worry when earth spin itself is 1% of static effect, which is 1000 times at-least, smaller, than 1 nanosecond.

In other words, Sun’s effect, will be slightly more or less, between earth and satellite, by 10^-14, a digit in the 14th figure. If we adjust our digits, it will come to order 13 or go to order 15. So, we can safely say, the effect of Sun because of earth and satellite orbiting about Sun is, < 10^-12 nano-second = 10^-21 seconds. Anything at 10^-8 seconds is important.

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How time dilation on GPS clocks is induced by static and spinning earth observations.

The image above shows earth at an altitude of 20200 + 9.08 km. The red line I drew is a straight line of 696 km, 3-D straight line — perhaps non-Euclidean, near the CERN – Gran Sasso Baseline, parallel to the latter, which is not straight and about 733 km long.

— 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 we┬áimagine 13.7 km/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 Gran-Sasso.

That is, 3.54375 E-7 rad / time-of-flight.

The earth spin is 3.000823 E-2 rad / time-of-flight — earth spin being 7.292 E-5 / s.

I have shown that static earth effect is less than 1 pico-sec — special-relativistic-effect is small and negative, compared to general-relativistic-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-inertial effect 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.

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OPERA neutrino anomaly, from Gravitational contribution.

This is original research work of Manmohan Dash, first published here on 6 November 2011. 12:13 pm.

an important update; OPERA neutrino anomaly, from Gravitational contribution.

Update, correction;

Here we will have to multiply the Gravitational shift by a 19.758 (4.4451*4.4451), as the 4.44 went to the denominator, instead of the numerator. A funny mistake, since quasi-consciously I left the E-10 factor, where it belonged, in the numerator, since at some level, I realized, I am doing some mistake otherwise.

One can see it’s not easy to guess, but one needs a pen and paper. Luckily that quasi-conscious thing, didn’t change the order of time shift and we are quite close to correct answer.

So all-in-all 0.5468 fs >> 10.804 fs.

The fractional shift in time, is not an inverse of fractional shift in frequency — as I stated or intended to calculate, it is same as the fractional shift in frequency, exact to the order of smallness, of the fractional shift of frequency itself. Since that factor is very small in our case, the fractional shift in time, can be set equal, to a fractional shift in frequency — a very important point.

The above update is summarized, in all detail, in the following 4 scanned pages:

Refer this:

I had made a quick calculation, of how relative velocity of GPS satellite, of ~4 km / s wrt the baseline — it has to be transverse, hence a transverse Doppler effect was actually calculated, introduces a time dilation of 0.216 ps — that is ~0.25 ps. But this was wrt the time of flight of 2.43 milli-secs — 730 kms at speed of light, in that case this becomes 88.89 ps for 1 second of time-of-flight. That is, if the satellite were at the level of moon, it would introduce, that kind of fractional time dilation, to processes measured on the baseline, on earth.

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Effect of Sun’s gravity on time dilation of atomic clocks.

This is an original research work of Manmohan Dash, published here, 3rd November 2011, 8:10 am.

What are the effect of Sun’s gravity, on time dilation, of atomic clocks, on GPS satellite and on on time dilation on clocks on earth.

GPS Satellite; revolves every 12 hrs around earth. Above ~26000 kms. Keeps time on earth driven by atomic clocks. Atomic clocks would lose time due to energy loss in gravitational field, its called as Gravitational time dilation — which would be red shift if time is gained in recession from the trap, and blue shift if entering into the trap ! So the idea is how much time is lost due to Sun’s presence or say Jupiter’s presence?

Apart from Sun, the other planets in the solar system, can exert similarly understood time dilation effect, although there will be no symmetry in such situations, but for a 24 hrs period, that concerns the GPS satellite, because thats –twice, their time period, for one single rotation around earth, this fact of non-symmetry may not matter.

The masses of other planets in the solar system, vis a vis time dilation effects, will be far smaller eg compared to that of Sun’s mass, hence their effects as well, considering, their distances also play a role. Although we can take only the massive and closer planets, or make a table of these parameter of such planets

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