GPS Time

[Little Bang]

Can someone tell me the difference in time on a GPS clock than the same clock on the ground?

[UncleAl]

Versus a ground observer, the three GPS atomic clocks/satellite run slower for their relative velocity (Special Relaltivity) and faster for being higher out of the gravitational well (General Relativity).

 

What the Global Positioning System Tells Us About Relativity

 

http://tycho.usno.navy.mil/ptti/ptti2002/paper20.pdf

http://www.eftaylor.com/pub/projecta.pdf

http://www.public.asu.edu/~rjjacob/Lecture16.pdf

Relativity in the Global Positioning System

Relativity in the GPS system

 

Relativity in the Global Positioning System

Physics of GPS relativistic time delay « Unused Cycles

Relativistic effects on orbital clocks

[Little Bang]

Ok Uncle, how about a geosynchronous satellite?

[CraigD]

We’ve discussed calculating the time dilation of satellites in circular orbits at least a few times in the past few years (eg: in 5/2008’s Re: An exercise the application of relativity to satellites – and by “we”, I see it’s actually you and me, LB). The gist of it is, the higher your orbit, the faster your clock.

 

The exact formulae are:

[math]v= \sqrt{\frac{u}{r}}[/math]

[math]\frac{t_s}{t_f} = \sqrt{1-\frac{v^2}{c^2}} = \sqrt{1-\frac{u}{c^2r}}[/math]

[math]\frac{t_s}{t_d} = \sqrt{1-\frac{2u}{c^2r}}[/math]

where: [math]u = G M[/math], a constant, Earth’s (or any other body you care to orbit) standard gravitational parameter;

[math]v[/math] is a satellite in circular orbit’s speed;

[math]t_s[/math] is the elapsed time of the satellite;

[math]t_f[/math] is the elapsed time of an observer at rest relative to the Earth; and

[math]f_d[/math] is the elapsed time of an observer very distant from the Earth.

 

Putting together the two kinds of time dilation gives total time dilation, and comparing it to a point on the Earth ([math]r \dot= 6578 \,\mbox{km}[/math], stationary at the poles, to about 478 m/s at the equator), you can find the time dilation for a satellite in a circular orbit at any altitude.

… how about a geosynchronous satellite?

Using the above formulae, at an geosynchronous altitude of about 42164 km, the time dilation relative to a point on Earth is between about 1.00000000053899601 and 1.00000000053772289, compared to between 1.00000000042704857 and 1.00000000042577546 for one at typical GPS satellite altitude of about 20200 km. For an observer very far from and at rest relative to the Earth, the factor’s about 1.00000000067548042 to 1.00000000067420731.

 

It’s interesting to note that at an altitude between about 3270 and 3289 km, a satellite’s clock is synchronized with one on a particular place on Earth’s surface, while at altitude’s below that, the factor is less than 1 (about 0.99999999969400984 to .99999999969273673 at 200 km, about as low as you can get without hitting too much atmosphere), that is, clocks in low Earth orbit run slower than clocks on the surface.

 

A warning about the “abouts” in the above: despite their apparent precision, they were calculated using only the Earth, ignoring the Sun and other bodies, so really are only approximate. Time dilation due to the Sun is greater on the scale of the orbit of planets is greater than due to the Earth on the scale of typical Earth satellite orbits – for example, Mercury’s orbit has a time dilation factor relative to Earth orbit of about .999999961751477, Jupiter’s about 1.000000011960276, about 100 times greater than the preceding Earth orbit examples.