GPS Clock After One Year: Time & Relativistic Effects

[SciencePF]

Suppose i have two clocks on Earth showing 12:00 hours.
After, one of the clocks is put in a gps satellite for a year.
After one year that clock returns to earth.
Due to relativistic effect what time that clock shows? Less than 12:00?
If so why?

 

[Colin1]

A GPS satellite IS a clock, isn't it? A fair few communications systems rely on them for sync. So is the question

If a terrestrial system was taking its clocking from a GPS satellite and you could bring the clock source to Earth would there be a relativistic deviation in sync between master and slave?

The problem with just two disparate clocks is that they're independent of one another, one is not a sync source for the other so I'm not sure anything gainful could be deduced from the experiment.

 

[russ_watters]

I'm not really sure what you mean, Colin1 - GPS clocks are synchronized to ground stations and are used to provide accurate time signals for various purposes including but not limited to navigation. The gps signal itself is a coded time signal.

GPS clocks are preset before launch to run at a rate that will enable them to stay roughly synchronized with ground stations. Anyway, that correction is...

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion.

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativitic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)!

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

So, if GPS satellietes didn't have their rates altered, they would gain 38 microseconds a day. An uncorrected clock in a satellite brought back to Earth a year later would show (at noon) 12:00:13.87 seconds. It would be almost 14 seconds fast!

 

[Sorry!]

So in reality when my physics teacher uses the example of astronaughts going to space and coming back aging slower than us on Earth really isn't true because of general relativity and they would actually age faster... or do they reach speeds fast enough to compensate for this?

 

[yogi]

No - the physics teacher is right - it is just in the case of GPS that the altitude effect is greater than the velocity effect - if the clock is not in orbit but traveling at half the speed of light for a long period, the travling clock would show less time

 

[russ_watters]

GPS Satellites orbit unusually high - 20,000 km - and slow - 14,000 km/h - compared with 240 km and 27,000 km/h for most normal satellites and real astronauts. I suspect, but don't really know, that the SR correction would be the bigger factor for astronauts. They'd age slower.

Yogi, I'm not sure if Sorry!'s teacher was talking about real astronauts or the hypothetical ones from the twins paradox...

 

[SciencePF]

Ok! Now if the clock that stay on earth, after a year, is put on the gps satellite what time did a eventual gps being reads when it arrives there?

 

[russ_watters]

That's a jumbled mess of an attempted sentence. Are you asking how a ground-based clock would compare to an uncorrected GPS clock when sent into orbit after a year on the ground? It would read 14 seconds slower than the GPS clock - same answer as the previous question (for our purposes it's really the same question).

 

[Jorrie]

GPS clock correction

GPS Satellites orbit unusually high - 20,000 km - and slow - 14,000 km/h - compared with 240 km and 27,000 km/h for most normal satellites and real astronauts. I suspect, but don't really know, that the SR correction would be the bigger factor for astronauts. They'd age slower.

You're right. I calculated that a 240 km altitude satellite clock loses about 10 ms/year relative to a ground clock.

BTW, I think you made a typo in a previous post - an uncorrected GPS satellite clock would gain about 14 ms/year relative to a ground clock, not 14 seconds.

 

[yogi]

When a clock is in orbit, the slow-down due to velocity is always going to be less than the greater rate required to adjust for the height assuming the reference is taken as the non-moving Earth centered frame. One could propose an object circling the Earth at o.5 c which results in a greater loss of time due to velocity than the increase required for the gravitational potential adjusment - but such an object is not in orbit - the frame of the object is no longer a free fall inertial frame - we are back to a traveling twin type of analysis or some such thing

 

[Jorrie]

When a clock is in orbit, the slow-down due to velocity is always going to be less than the greater rate required to adjust for the height assuming the reference is taken as the non-moving Earth centered frame.

Agreed, but do you agree that for a LEO satellite, the slowdown due to velocity dominates when referenced to a terrestrial clock? I find that to be the case below about 3200 km altitude, where the velocity and gravitational effects cancel out.

 

[Sorry!]

GPS Satellites orbit unusually high - 20,000 km - and slow - 14,000 km/h - compared with 240 km and 27,000 km/h for most normal satellites and real astronauts. I suspect, but don't really know, that the SR correction would be the bigger factor for astronauts. They'd age slower.

Yogi, I'm not sure if Sorry!'s teacher was talking about real astronauts or the hypothetical ones from the twins paradox...

yeah it's real astronaughts :p thanks for the answers :)

 

[russ_watters]

When a clock is in orbit, the slow-down due to velocity is always going to be less than the greater rate required to adjust for the height assuming the reference is taken as the non-moving Earth centered frame.

I wouldn't say "always" - there is an orbital velocity that corresponds to sea level, afterall. So the question is, is the altitude below which the SR correction is bigger than the GR correction high enough to be above the atmosphere? Jorrie seems to have calculated that that altitude is somewhere above LEO.

 

[russ_watters]

BTW, I think you made a typo in a previous post - an uncorrected GPS satellite clock would gain about 14 ms/year relative to a ground clock, not 14 seconds.

You're right - I confused my prefixes (micro with mili). I thought 14 seconds seemed awfully large.

 

[yogi]

I wouldn't say "always" - there is an orbital velocity that corresponds to sea level, afterall. So the question is, is the altitude below which the SR correction is bigger than the GR correction high enough to be above the atmosphere? Jorrie seems to have calculated that that altitude is somewhere above LEO.

Correct Russ - if you reference all clocks to an equipotential defined by sea level, then there is no correction for height for an object traveling in an evacuated tunnel at orbit velocity - I had in mind a different reference point.

 

[Jorrie]

Ground-orbit time equality

Jorrie seems to have calculated that that altitude is somewhere above LEO.

Here's my calculation. I think the simplest way to compare clocks is to express both relative to the time (t) of a distant static observer in asymptotically flat spacetime, considering Earth as an isolated body at rest at the origin (i.e. Schwarzschild coordinates).

On Earth's surface, radius R and taking the rotation speed as insignificant, the clock rate relative to t is:

d\tau_0/dt \approx \sqrt{1-2GM/(R c^2)}

For a satellite orbiting at constant radius r, the clock rate relative to t is:

d\tau_1/dt = \sqrt{1-2GM/(r c^2)-v_o^2/c^2} = \sqrt{1-3GM/(r c^2)},

since circular orbital velocity v_o^2 = GM/r.

It is now easy to spot that d\tau_0/dt = d\tau_1/dt when r = 1.5R, giving an altitude of ~3190 km above MSL. Below this altitude uncorrected satellite clocks lose time and above it they gain time relative to a terrestrial clock.

 

[Jorrie]

It is now easy to spot that d\tau_0/dt = d\tau_1/dt when r = 1.5R, giving an altitude of ~3190 km above MSL. Below this altitude uncorrected satellite clocks lose time and above it they gain time relative to a terrestrial clock.

If the tangent velocity for an equatorial clock (v \approx 463 m/s due to Earth's rotation) is taken into account, the altitude for ground-orbit clock rate equality is about 3170 km, from:

1 - \frac{2GM}{Rc^2} - \frac{v^2}{c^2} =1 - \frac{3GM}{rc^2}

giving

r=\frac{3GMR}{2GM+Rv^2}

 

[plbeale]

Has anyone ever brought a clock back from orbit and compared it with a clock on the ground?

 

[russ_watters]

I don't think so -- but why would that matter?

 

[SciencePF]

if when they are both on the Earth surface they read p.e. 12 oclock, and then one is put in orbit and after a certain period it came to Earth surface did they show the same number (time)?

 

[FrankMak]

GPS clocks do not show time-of-day, they provide a precise cyclic rate.

"The U.S. Naval Observatory (USNO) monitors the timing of the GPS to provide a reliable and stable coordinated time reference for the satellite navigation system."

U.S. Naval Observatory

If you brought a GPS clock to earth, it would have a cyclic rate that is different from the earth-based GPS reference. It was launched with this difference to compensate for the orbital altitude difference.

"To account for this, the frequency standard onboard each satellite is given a rate offset prior to launch, making it run slightly more slowly than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz."

http://www.gpsfleetsolutions.com/GPS_Technology_Technical.php

 

[Mike Holland]

Found this on WIKI, and thought it would be of interest.

"The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity."

Mike

 

[Dickfore]

GPS Satellites orbit unusually high - 20,000 km - and slow - 14,000 km/h - compared with 240 km and 27,000 km/h for most normal satellites and real astronauts.

Please provide a source to justify your claim.

 

[russ_watters]

The reference is in the first post i made...5 years ago. Couldn't you have googled that or checked my reference in the meantime! Sheesh.

 

[Dickfore]

The reference is in the first post i made...5 years ago. Couldn't you have googled that or checked my reference in the meantime! Sheesh.

I found the following info:
http://tycho.usno.navy.mil/gpsinfo.html#seg

The SPACE segment consists of 24 operational satellites in six orbital planes (four satellites in each plane). The satellites operate in circular 20,200 km (10,900 nm) orbits at an inclination angle of 55 degrees and with a 12-hour period. The position is therefore the same at the same sidereal time each day, i.e. the satellites appear 4 minutes earlier each day.

So, I guess they are pretty high.

 

[pervect]

GPS clocks do not show time-of-day, they provide a precise cyclic rate.

"The U.S. Naval Observatory (USNO) monitors the timing of the GPS to provide a reliable and stable coordinated time reference for the satellite navigation system."

U.S. Naval Observatory

If you brought a GPS clock to earth, it would have a cyclic rate that is different from the earth-based GPS reference. It was launched with this difference to compensate for the orbital altitude difference.

"To account for this, the frequency standard onboard each satellite is given a rate offset prior to launch, making it run slightly more slowly than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz."

http://www.gpsfleetsolutions.com/GPS_Technology_Technical.php

There's more to it than that. GPS time is deliberately steered to agree with UTC time, rather than to maintain a precise cyclic rate.

See for instance
http://tycho.usno.navy.mil/gpstt.html

GPS TIME STEERING
GPS time is automatically steered to UTC(USNO) on a daily basis to keep system time within one microsecond of UTC(USNO), but during the last several years has been within a few hundred nanoseconds. The rate of steer being applied is +/-1.0E-19 seconds per second squared.

Thus there is more to GPS time than a precise cyclic rate - it is in fact steered towards UTC time.

http://tycho.usno.navy.mil/ptti/1986/Vol%2018_20.pdf clarifes a point about this which bothered me. It notes that GPS time is steered towards UTC time after leap seconds have been removed from UTC time. I would think that this could be more simply stated as GPS time is steered towards TAI time (because TAI time, according to my understanding, doesn't have the leap seconds), but that's not they way that USNO phrased it.

http://en.wikipedia.org/wiki/International_Atomic_Time

nternational Atomic Time (TAI, from the French name Temps atomique international)[1] is a high-precision atomic coordinate[2] time standard based on the notional passage of proper time on Earth's geoid. It is the basis for Coordinated Universal Time (UTC), which is used for civil timekeeping all over the Earth's surface, and for Terrestrial Time, which is used for astronomical calculations. Since 30 June 2012 when the last leap second was added,[3] TAI has been exactly 35 seconds ahead of UTC. The 35 seconds results from the initial difference of 10 seconds at the start of 1972, plus 25 leap seconds in UTC since 1972.

GPS time is, like TAI time and UTC time, is a coordinate time standard. This means that the proper time interval that you'd actually measure with a local atomic clock would need to be adjusted for your particular location (mainly, it'd havea to be adjusted for the altitude), as the difference in coordinate times is equal to the proper time only on the Earth's geoid (basically, at sea level).

 

[FrankMak]

"GPS clocks do not show time-of-day, they provide a precise cyclic rate."

The Cesium clocks provide a constant cyclic rate. If it is necessary to provide a time-of-day display someplace, I am sure this is handled in the software.

Please note the following term in the URL below, "user must use the correction terms".

IS-GPS-705 PIRN-004

Note they have updated the Frequency Plan 3.3.1.1 to 10.2299999954326 MHz.

Earlier reports stated the offset is hard coded before launch. Certainly wouldn't want to leave a software door open so somebody could hack the offset.

See section GPS time and date in URL below about TAI time:

http://www.gpsfleetsolutions.com/GPS_Technology_Technical.php

 

[FrankMak]

Mike,

Have you read a rebuttal of the Hafele–Keating experiment?

Hafele and Keating Travel Around the World

H&K Paper 1971

I did not see any information on the characteristics of the platforms, the aircraft used, and the inflight environment. What environmental sensors were used, and how was the environmental data recorded and correlated with the Cesium clock data?

The Maryland Experiment was better documented, but not completely.

MarylandExperiment

They should have measured permittivity to determine whether it changed with altitude.