Relativity correction to GPS
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Exercise:
GPS satellites move at about vO. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to one part in ppmO.
Solution:
SolQtyp/ppmX SolQtygtvX^/nccX^ The Lorentz factor corresponding to the velocity of a GPS satellite is: gamma fracsqrt-fracv^c^ g && textnot accurate enough &approx -fracfracv^c^ && textTaylor &approx -frac gt gamma - dg If we denot the time passed on Earth's atomic clock with t and the time passed on the satellites atomic clock with t_ the time error relative to the time passed on the satellites clock is: fract-t_t_ fracgamma t_-t_t_ gamma - dg fracppm p Time dilation if not accounted for would roduce an error of about part in numpre which is numpr times greater than the precision of atomic clocks. Not correcting for time dilation means a receiver could give a much poorer position accuracy. GPS devices must make other corrections as well including effects associated with General Relativity. This effect is even bigger. medskip textbfApproximation with Taylor expansion: The binomial expansion of pm x^n approx mp nx for xll . With n-frac we get -x^-frac -frac x.
GPS satellites move at about vO. Show that a good GPS receiver needs to correct for time dilation if it is to produce results consistent with atomic clocks accurate to one part in ppmO.
Solution:
SolQtyp/ppmX SolQtygtvX^/nccX^ The Lorentz factor corresponding to the velocity of a GPS satellite is: gamma fracsqrt-fracv^c^ g && textnot accurate enough &approx -fracfracv^c^ && textTaylor &approx -frac gt gamma - dg If we denot the time passed on Earth's atomic clock with t and the time passed on the satellites atomic clock with t_ the time error relative to the time passed on the satellites clock is: fract-t_t_ fracgamma t_-t_t_ gamma - dg fracppm p Time dilation if not accounted for would roduce an error of about part in numpre which is numpr times greater than the precision of atomic clocks. Not correcting for time dilation means a receiver could give a much poorer position accuracy. GPS devices must make other corrections as well including effects associated with General Relativity. This effect is even bigger. medskip textbfApproximation with Taylor expansion: The binomial expansion of pm x^n approx mp nx for xll . With n-frac we get -x^-frac -frac x.