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38,000 nanoseconds = 38,000 feet = 7 miles error per day if the GPS was not corrected for special and general relativity.
This agrees with a simple analysis based on nothing more than the orbital radius of GPS satellites:
Because the orbital radius is r = 26,600 km (≅ 4.17 R_e), the orbital speed is v = sqrt(GM_e/r) = 3.87 km/s.
The special relativistic effect causes GPS clocks to run SLOW by gamma – 1 ≅ (1/2)v^2/c^2 = (1/2)GM_e/rc^2 = 7.2 microsec/day.
The general relativistic effect causes GPS clocks to run FAST by delta phi/c^2 = GM_e/rc^2(r/R_e – 1) = 45.6 microsec/day.
Thus, the overall effect is that GPS clocks run fast by about 38 microsec/day. It's interesting to note that the general relativistic effect is by far the largest contributor and it's also interesting to note that the orbital radius at which the two effects cancel each other is r = 3/2 R_e.
Cal Poly Pomona
On Oct 30, 2010, at 1:53 PM, Bernard Cleyet wrote:
38 microseconds – discrepancy in GPS satellite time per day (compensated by clock speed) due to relativity
Microsecond - Wikipedia, the free encyclopedia
bc finds the darndest things while googling.