Re: GPS calculations
From: Henri Wilson (H_at_..(Henri)
Date: 09/29/04
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Date: Wed, 29 Sep 2004 22:44:38 GMT
On Wed, 29 Sep 2004 06:32:58 -0400, "jahn" <suzysewnshow@yahoo.com.au> wrote:
>
>"Henri Wilson" <H@..> wrote in message news:3aojl0d0l2usb2s61je64lplrgbkhvtlrh@4ax.com...
>> On Tue, 28 Sep 2004 14:49:01 -0500, EjP <nospam@hackers.are.bad> wrote:
>>
>> >xxein wrote:
>> >> For those confused by 46 us/day and 38 us/day, It depends on wether
>> >> you think the satellites orbit once a day or twice a day.
>> >>
>> >> It seems there is a lot of confusion both on this ng and the general
>> >> science archives.
>> >>
>> >> End.
>> >
>> >There is no confusion amongst those who know what they're talking
>> >about. The correction is 38 usec a day, based on a perturbative
>> >treatment which allows you to separate the "SR" and "GR" compoenents.
>> >If you do this, you get a 45 usec advance from the gravitational
>> >potential difference and a 7 usec slowing form the SR component,
>> >giving a net correction of 38 usec. This was calculated *long*
>> >before the satellites were launched and match observation exactly.
>> >
>> >Since the orbital period of GPS satellites is about 12 hours,
>> >the correction per period is about 19 usec. I'm not sure
>> >where your 46 usec comes from, except that it's close to the
>> >45 that one gets from a partial calculation.
>> >
>> >The calculation is actually pretty simple. I've appended it
>> >below. This version is a bit long because I was arguing with
>> >some kook that claimed all sorts of "extra" factors had been
>> >added, and I was showing that you can calculate the whole thing
>> >if you know the altitude and period of the satellite and
>> >the radius of the Earth.
>> >
>> > -E
>> >
>> >
>> >===============Details of GPS Calculation========================
>> >The altitude of the GPS sats are about 20,000 km, so their
>> >radius is 27,000 km. The observed period is 12 hours. From this you
>> >can work out that the velocity is
>> >
>> > v = 2*PI*R/T = 3.93 x 10^3 m/s
>> >
>> >This means that \beta = v/c = 1.31 x 10^(-5). The velocity at the
>> >surface
>> >of the earth is negligible compared to this, so the SR clock correction
>> >would come from
>> >
>> > \gamma = 1/sqrt{1-\beta^2} ~ 1 + .5*\beta^2 ~ = 1 + (8.5 x 10^(-11))
>> >
>> >so the orbiting clock would tick "slowly" by 1/gamma, or a fractional
>> >change of
>> >
>> > 8.5 x 10^(-11)
>> >
>> >and the clock would lose (86400)*(8.5 x 10^(-11)) = 7 x 10^(-6) s/day
>> >
>> >
>> >Now, move on to GR. Use
>> >
>> > v = sqrt(GM/R) to get GM = 4.2 x 10^14
>> >
>> >The GR gravity-only time dilation "gamma" is (look it up)
>> >
>> > \gamma = 1/sqrt{1 - 2GM/(Rc^2)} ~ 1 + GM/(Rc^2)
>> >
>> >So the difference between a clock ticking on Earth (R=Re=6.7E6) and one
>> >ticking in orbit
>> >(R=Ro=27E6) would be
>> >
>> > 1+GM/(Re*c^2)
>> > ------------- ~ 1 + (GM/c^2)(1/Re - 1/Ro)
>> > 1+GM/(Ro*c^2)
>> >
>> >since the earth clock is deeper in the gravitational well, it would tick
>> >slowly and relative to it, the GPS clock would tick fast by a factor
>> >
>> > (GM/c^2)(1/Re - 1/Ro) = 5.23 x 10^(-10)
>> >
>> >so the clock would gain
>> >
>> > 86400*(5.23 x 10^(-10)) = 45 x 10^(-6) s/day
>> >
>> >so the net gain would be 45 - 7 = 38 microseconds, which is (strangely
>> >enough) EXACTLY the value they use, see
>> >http://www.phy.syr.edu/courses/PHY312.98Spring/projects/GPS/GPS.html
>> >
>>
>> So what is it about movement or gravity wells that actually affects the way a
>> clock ticks?
>>
>> You CDEFs all talk crap.
>>
>> Just because light increases speed and incurs a doppler shift as it falls down
>> a gravity well, you think 'clock ticks' do the same.
>>
>> Maybe PA's famous tick fairies are at it again....!
>Yikes!
>Do moving clocks tick slower?
>Hint: Motion is relative.
><< Numerically integrating the satellite
>motion equations relative to the center of the earth is done for
>numerical reasons,where the equations that are integrated are
>the formal difference (not a Lorentz transformation) of the
>equations of motion of the GPS satellite and earth in the
>barycentric frame.>>
>http://www.draper.com/publications/digest03/paper203.pdf
>
>Do photons Fall?
>Hints:
>You can't see an event before it happens.
>The values of mu and epsilon for all known materials is greater than the value in a vacuum, so the speed of light has a maximum
>value in a vacuum.
>Are atomic clock affected by gravity?
>Hint: The accuracy of Earth-bound cesium atomic clocks is limited by the length of time each cesium atom can be observed -- on
>Earth, gravity quickly removes the atoms from the observation region. In the microgravity environment of space, each atom can be
>observed for many seconds.
>
>
>Kind regards,
>Sue...
I cannot understand what you are getting at.
....and please adjust your line length.
>>
>>
>> HW.
>>
>> www.users.bigpond.com/hewn/index.htm
HW.
www.users.bigpond.com/hewn/index.htm
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