Yes, it has to do with the ratio of orbital velocity versus proximity to the Earth's center of mass.

The ephemeris data should be available for the ISS and any GPS SV on the web (we use the USAF database for the GPS SVs), but the long and short of it is:

GPS SV Altitude above geoid = ~20,000 km

GPS SV Orbital Velocity = ~14,000 km/hr

ISS Altitude above geoid = 330 km

ISS Orbital velocity = 27,744 km/hr

While the velocity is nearly 2X for the ISS, the altitude above the geoid is 60 times lower than the GPS SV.

I don't have time to plug in the numbers, but the two different orbital parameters for the two SVs demonstrates the effects of special and general relativity and their dominance for those parameters.

So, Special Relativity is concerned with the relative velocities of two observers (i.e., Earth and the SV). General relativity is concerned with the proximity to Earth and the effect of gravity.

The faster we go, the slower the clock gets compared to a "stationary" observer's clock. The closer the clock is to the Earth, the slower the clock runs.

So, the GPS orbital altitude dominates over the Special Relativity effects of the SV's velocity. The converse is true for the ISS, where proximity to Earth combines with Special Relativity's effects to slow the clock down.

Hi Hero, good answer.

You concluded with: "So, the GPS orbital altitude dominates over the Special Relativity effects of the SV's velocity. The converse is true for the ISS, where proximity to Earth combines with Special Relativity's effects to slow the clock down."

The final sentence may be a bit confusing to others, because it may be interpreted that both velocity and gravity slows the ISS clocks down relative to the ground.

Any guess on the altitude for earth-orbit clock rate equality?

Jorrie

Essentially, setting the equation for general relativity equal to the equation for special relativity and solve for v should give you the orbiting velocity where they cancel out.

sqrt(1-(2GM/Rc^2)) = sqrt(1-v^2/c^2)

However, doesn't that yield the escape velocity of the Earth (11.2 km/s)? So I don't think there is an orbit where this happens, right?

Hi Hero.

No, your equation is not quite applicable. One must find what happens at geoid level (~terrestrial clocks) and what happens at orbit level clocks, each with height and velocity effects and then equate the equations for the clocks.

Jorrie

Opps. I forgot to normalize for the reference point!

I also did not consider the Earth's rotation, which adds Special Relativity effects of about 138 ns per day (clock runs slower by 138 ns).

Hero, you wrote: "Opps. I forgot to normalize for the reference point!"

The simplest reference point to use is, interestingly enough, a distant observer at rest with respect to Earth. Then the equations become simple Schwarzschild and one can easily equate them and solve for radial distance to obtain height.

Jorrie

I tried that and I think I did something wrong with the equation because the numbers I got were 4 to 5 times larger than expected. Not sure what I did wrong.

If I get a little more time I will try to play with it tomorrow.

Hi Hero.

One don't need the full relativistic equations for the low gravity and low speed orbits around Earth. A nice simplifying approximation is this:

dτ₁/dt ≈ 1 - GM/r₁ - ½v₁²/c² = dτ₂/dt = 1 - GM/r₂ - ½v₂²/c²,

where dτ is the time interval near Earth and dt is the time interval for a distant static observer.

Multiply both sides by 2c², cancel out and we have:

2GMc²/r₁ + v₁² = 2GMc²/r₂ + v₂²

For Earth, GMc² = 3.48 x 10^31. Take r₂ as Earth's radius and v₂ as the equatorial rotational speed. Now the trick required is to solve the left hand side for r₁. (I'll leave a little challenge).

Jorrie

PS: it seems that few engineers have the time and energy to struggle with this 'esoteric' stuff. We are a down-to-earth lot, it seems. Look at the response to the floor tiles challenge...

I wrote: "One don't need the full relativistic equations for the low gravity and low speed orbits around Earth. A nice simplifying approximation is this:

dτ₁/dt ≈ 1 - GM/r₁ - ½v₁²/c² = dτ₂/dt = 1 - GM/r₂ - ½v₂²/c²,

where dτ is the time interval near Earth and dt is the time interval for a distant static observer.

Multiply both sides by 2c², cancel out and we have:

2GMc² /r₁ + v₁² = 2GMc²/r₂ + v₂²"

Those c²'s were actually erroneous 'red herrings' and should be ignored (a dimensionality analysis shows up the error!). The equations should simply be:

dτ₁/dt ≈ 1 - GM/r₁ - ½v₁² = dτ₂/dt = 1 - GM/r₂ - ½v₂²

and

2GM/r₁ + v₁² = 2GM/r₂ + v₂²,

because GM/r already has the dimension m²/s².

The final 'trick' is to realize that v₁² is the orbital velocity squared: v_o² = GM/r₁, giving:

3GM/r₁ = 2GM/r₂ + v₂²

from which is easy to see that the required radius from Earth's centre of mass (c.o.m.) is:

r₁ = 3GM/(2GM/r₂ + v₂²)

where v₂ is the surface velocity due to Earth's rotation at the location of the clock and r₂ is the distance of that clock from Earth's c.o.m.

Jorrie

I think my numbers are right!

Altitude = 3189 km

Velocity = 6456 m/s

That would be a circular orbit.

Yep, close enough! (It seems we happened to post more or less at the same time.)

Did you consider the Earth clock's movement due to the rotation of Earth?

I got some 3170 km for an equatorial clock, but it depends a bit on what geoid radius one assumes and so on.

Let's see if someone else bites on this cherry...

Jorrie

"PS: it seems that few engineers have the time and energy to struggle with this 'esoteric' stuff. We are a down-to-earth lot, it seems. Look at the response to the floor tiles challenge...

That's too funny!

I did consider both GR and SR for all positions. After my first effort to try to solve for both equations equal to zero in a previous post, I then did this in a hack way and I just regrouped and slogged through it bit by bit to make sure I wasn't missing something. In the end what I did was created a spreadsheet in Excel and the resulting columns were altitude and the sum of the GR and SR effects. I could see where the convergence was pretty easily once I found the error of my ways with the GR calculation. I am a visual type of guy.

I really need to improve my math skills. Unfortunately, I lost all my first year college text books for physics and calculus. I saved all my other books for the remaining years, but switched my math/physics major to premed, then at the very end of that (after a 4.0 average) I just punted and joined the workforce in the engineering field! ;-)

I keep telling myself that when things slow down a little I will pick up some of the more advanced math studies to round out my knowledge a little more. Then again, there are so many other subjects of interest that compete that it is hard to get anything done!