The Global Positioning System (GPS) features 24 earth satellites orbiting at altitude \(20,200 \mathrm{~km}\) above earth's surface. Each satellite carries 4 highly precise atomic clocks; this precision is essential in allowing us to know our positions on the ground within a few meters or less. Special relativistic time dilation effects, although tiny, must be taken into account. They are due to the speed \(v\) of the satellites relative to a clock at rest in some appropriate inertial frame. Let us take this reference clock to be a hypothetical clock at rest at the center of the earth. (To call such a clock inertial is only an approximation, because the earth has a small acceleration toward the sun and moon, which are themselves accelerating toward the center of our galaxy, etc., etc.)

(a) Find the special-relativistic time dilation factor \(\sqrt{1-v^{2} / c^{2}}\) for clocks in a GPS satellite, expressed in the form \(1-\epsilon\), where \(\epsilon\) is a very small number.

(b) How much time would they lose in one year due to this effect? (There is a second relativistic effect on GPS clocks, as described in Chapter 10, due not to their velocity but to their altitude in earth's gravity. Given information: Mass and mean radius of the Earth: \(5.98 \times 10^{24} \mathrm{~kg}\) and \(6370 \mathrm{~km}\); Newton's gravitational constant \(\left.G=6.674 \times 10^{-11} \mathrm{~m}^{3} /\left(\mathrm{kg} \mathrm{s}^{2} \right).\right)\)