Atomic Clock in Your Pocket
In “GPS and Time Dilation” we discussed how your smart phone uses GPS to identify your location, and we saw that an error in time measurement, even as small as a billionth of a second, is intolerable:
But if on GPS satellites clocks are atomic (and they are as big as in the image above), your smart phone clock is cheap and small. Then how can signal delay time be used at all, if your time cannot be measured so precisely? It turns out, clock of the receiver is not used for triangulation, but your smart phone can still get local time as precisely as GPS satellites get their time from their atomic clocks. For this, extra one or two satellites are used in addition to the three satellites required for the triangulation (of the 24 GPS satellites on the orbit). Let’s consider two satellites S1 and S2, which broadcast their known coordinates (X1, Y1, Z1) and (X2, Y2, Z2), and their known time T1 and T2. And your receiver/smart phone R is ignorant about its own position (X, Y, Z) and its local time T. Distances R1 and R2 from the receiver to the satellites are unknown as well, but there are two known formulas for the distance: one is by time taken for the signal to reach the receiver multiplied by c — the speed of light, another is by Pythagorean formula, which uses coordinates. Thus, we have following equations:
To get rid of T², X², … let’s subtract last two formulas from each other:
R1²-R2² = c²×(-2T1+2T2)×T + c²×(T1²-T2²) = (-2X1+2X2)×X + (-2Y1+2Y2)×Y + (-2Z1+2Z2)×Z + X1²+Y1²+Z1²-X2²-Y2²-Z2²
Thus,
c²×(-2T1+2T2)×T + (2X1–2X2)×X+(2Y1–2Y2)×Y + (2Z1–2Z2)×Z + c²×(T1²-T2²)-X1²-Y1²-Z1²+X2²+Y2²+Z2²=0
Since only T, X, Y, Z are unknown, let’s rewrite the above equation simpler (using Known* instead of real numbers):
Known1×T+Known2×X+Known3×Y+Known4×Z+Known5=0
We got a linear equation with 4 unknowns, using data broadcasted by satellites S1 and S2.
Absolutely the same way we can get another linear equation from satellites S1 and S3 data, and another from S1 and S4 data. Let’s for a moment use 5th satellite data as well, thus we will have 4 linear equations with 4 variables — that is enough for the receiver’s computer to solve 4 liner equations and find T, X, Y, Z. We used extra 5th satellite data/equation for simplicity — to stay with linear equations only. Actually, 3 linear equations will do, not yet to solve, but to resolve unknown X, Y, Z via T. Then the computer can substitute X, Y, Z as T-expressions in the equation for R1² (in the image above) and find T from that equation.
Thus, your smart phone knows/calculates local time T as precisely as the GPS satellites know their local time via their atomic clocks.
Next 👉1.000000000445 Seconds Pass in GPS Satellite, in 1 Earth-Second.
