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why orbital motion of earth around sun does not matter to GPS

November 30, 2011

Mohan, mdashf

Because it is not only earth, the satelite 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, you sicking Indians, mere pyare bhaion aur behenon) Earth-satelite is 26000 kms = 0.000174 AU ~2 milliAU. The angular speed of earth and satelite due to Sun will be almost same. w=(r)^-1.5 = sqrt( 1/ rrr), what difference it makes if you change r by 0.0002 when r is 1, the result is 1 changes to 0.9997. SO if the ang. speed of earth is 1 the ang speed of satelite 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 satelite by 10^-14, a digit in the 14th figure. If you adjust your digits it will come to order 13 or go to order 15. SO you can safely say the effect of Sun because of earth and satelite orbiting about Sun is < 10^-12 nano-second = 10^-21 seconds. Anything at 10^-8 seconds is important.

we knew this distance very closely even in 1672, Cassini computed Sun-earth to be 14 Karod kms = 140 million kms, the truer as we know today is 15 karod kms = 150 million kms.

Tyco Brahe measured: 8 miilion kms, Keppeler 24 million kms but Cassini was approximately correct 140 million kms,

note that: w=(GM_sun)^0.5/r^1.5 = sqrt(GM_sun / rrr). so v^2=v*v= GM_sun/r. where M_sun is large but r is 1 AU. v*v is what gives the order of time which is 1/r, so r=1 and r=0.99983 have a difference of 0.00017. That is the first change in time appears in the 4th decimal place .. between earth and satelite. You have to put the ang_speed due to orbit around Sun, in the right order before you can see the order of time (so I am reviewing the above, but seems if not 10^-21 it is at-least 10^-16 and only 10^-8 seconds matter)

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