Relativity Theory: Satellite Time Loss/Gain in Orbit

[Bjarne]

I want to know how much time (seconds per year) satellites are losing or gaining per year due to SR and GR.
Is there any difference whether the satellites are orbiting clockwise or anti clockwise?

 

[Drakkith]

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

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)!

 

[Bjarne]

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

Do you know where to read about if this is confirmed, and fits with the predictions ?

 

[Drakkith]

Do you know where to read about if this is confirmed, and fits with the predictions ?

Nowhere specific no. I'm pretty certain they are correct though. I've read about having to correct for relativity effects in the doc from here:

http://www.losangeles.af.mil/shared/media/document/AFD-100302-043.doc

This is from the air force site on GPS. Its a .mil site, which only the us military has. Its the real deal.

 

[russ_watters]

Do you know where to read about if this is confirmed, and fits with the predictions ?

The confirmation comes in the fact that the GPS system works as designed. Beyond that, I'm sure the service that maintains the GPS system keeps track of clock errors for maintenance reasons and would notice if Relativity was wrong, but I don't think they'd see a value in publishing that data to prove something that is already well proven and accepted.

 

[George Jones]

Do you know where to read about if this is confirmed, and fits with the predictions ?

Everytime GPS is used is a test that confirms it [edit]as Russ has already noted[/edit]. For a calculation that doesn't split the effect into contributions from special and general relativity, see

https://www.physicsforums.com/showthread.php?p=731738#post731738.

 

[Bjarne]

Everytime GPS is used is a test that confirms it [edit]as Russ has already noted[/edit]. For a calculation that doesn't split the effect into contributions from special and general relativity, see

https://www.physicsforums.com/showthread.php?p=731738#post731738.

Yes this is true, and For example Mercury (etc..) too.
But according classic orbit mechanics an ellipse orbit could not be maintained with any kind of resistance.

For example Mercury "should fall" when it do not reach the speed is must (by perihelion) , I guess nothing to Mercury, - but it “should", - if the correct speed always not is possible to be achieved.

I just wonder if there are any simply way to understand why Mercury (etc..) can "slow down" without any consequences.
What is the simple secret ?

I understand the KE balance, - but that too is affected by too slow speed ,
I am not sure if this matter, - but the speed certainly seems to be a problem...at least to understand what is the consequence, and how can Mercury survey it ?

Edit..
Let us say Einstein and Newton both was living at the space station, and both would drop a stone at the same time. – Einstein would know that 100 km/h + 100 km/h not is 200 km/h - but 199.999 …etc.. – if not extra energy was added. – Therefore Einstein would calculate more precisely when the stone would hid the Earth.

Or would Newton ?

 

[Bill_K]

These effects are pretty easy to calculate. It turns out they both depend on exactly the same quantity, v²/c² = GM/rc², where v is the orbital velocity. For the SR effect, the time slowdown is sqrt(1 - v²/c²) ~ - v²/2c². For the GR effect, the apparent speedup comes from the Schwarzschild metric ds² = (1-2GM/rc²) dt², which implies an approximate speedup ~ GM/rc².

On the face of it, both effects increase as 1/r as you get closer to Earth. However GR loses as you go down, because you need to compare the slowdown in orbit vs the slowdown on the ground, which modifies the answer by an extra factor Δr/r. GR is comparable for the GPS satellites because they are in a very high orbit: 20,000 km, or about three Earth radii. Low Earth orbit would mean an altitude about 50 times less than this, so GR will be 50 times smaller for your average satellite.

 

[Bjarne]

These effects are pretty easy to calculate. It turns out they both depend on exactly the same quantity, v²/c² = GM/rc², where v is the orbital velocity. For the SR effect, the time slowdown is sqrt(1 - v²/c²) ~ - v²/2c². For the GR effect, the apparent speedup comes from the Schwarzschild metric ds² = (1-2GM/rc²) dt², which implies an approximate speedup ~ GM/rc².

On the face of it, both effects increase as 1/r as you get closer to Earth. However GR loses as you go down, because you need to compare the slowdown in orbit vs the slowdown on the ground, which modifies the answer by an extra factor Δr/r. GR is comparable for the GPS satellites because they are in a very high orbit: 20,000 km, or about three Earth radii. Low Earth orbit would mean an altitude about 50 times less than this, so GR will be 50 times smaller for your average satellite.

This math is above my head, or maybe I am too lazzy

But I guess we can say:
Yes Einstein is right, the stone dropped from the space station will lose speed due to "relativistic resistance" and will in fact be delayed compared to what we would expect according to classic physics.

Mercury and satellites are also losing speed when approaching perihelion due to “relativistic resistance”.

BUT orbiting objects will also increase their mass due to speed, towards perihelion.

Right after reaching perihelion, > towards aphelion the increased (relativistic) mass means it is more difficult to decelerate (Centripetal Force).

Therefore all orbiting objects will get the "gravitational level” back again, due to the increased mass, because it is more difficult to slow down a heavier object on its way out of the gravitionel field.

In this way the "accumulated kinetic energy" (mass increase) is "released” , - not to higher speed , - but due to the simple fact that it is more difficult to get a heavy body to decelerate.

So due to the increased mass orbiting objects will reach the same "altitude" (“potential gravitionel energy level”) as from where these were starting.
I guess this must be the simplest way to understand it?