6 TOPEX/POSEIDON Relativity ExperimentRelativity in the Global Positioning 4 The Realization of Coordinate

5 Relativistic Effects on Satellite Clocks 

For atomic clocks in satellites, it is most convenient to consider the motions as they would be observed in the local ECI frame. Then the Sagnac effect becomes irrelevant. (The Sagnac effect on moving ground-based receivers must still be considered.) Gravitational frequency shifts and second-order Doppler shifts must be taken into account together. In this section I shall discuss in detail these two relativistic effects, using the expression for the elapsed coordinate time, Eq. (28Popup Equation). The term tex2html_wrap_inline1612 in Eq. (28Popup Equation) includes the scale correction needed in order to use clocks at rest on the earth's surface as references. The quadrupole contributes to tex2html_wrap_inline1612 in the term tex2html_wrap_inline1680 in Eq. (28Popup Equation); there it contributes a fractional rate correction of tex2html_wrap_inline1682 . This effect must be accounted for in the GPS. Also, V is the earth's gravitational potential at the satellite's position. Fortunately, the earth's quadrupole potential falls off very rapidly with distance, and up until very recently its effect on satellite vehicle (SV) clock frequency has been neglected. This will be discussed in a later section; for the present I only note that the effect of earth's quadrupole potential on SV clocks is only about one part in tex2html_wrap_inline1516, and I neglect it for the moment.

Satellite orbits.   Let us assume that the satellites move along Keplerian orbits. This is a good approximation for GPS satellites, but poor if the satellites are at low altitude. This assumption yields relations with which to simplify Eq. (28Popup Equation). Since the quadrupole (and higher multipole) parts of the earth's potential are neglected, in Eq. (28Popup Equation) the potential is tex2html_wrap_inline1688 . Then the expressions can be evaluated using what is known about the Newtonian orbital mechanics of the satellites. Denote the satellite's orbit semimajor axis by a and eccentricity by e . Then the solution of the orbital equations is as follows [13Jump To The Next Citation Point In The Article]: The distance r from the center of the earth to the satellite in ECI coordinates is

  equation286

The angle f, called the true anomaly, is measured from perigee along the orbit to the satellite's instantaneous position. The true anomaly can be calculated in terms of another quantity E called the eccentric anomaly, according to the relationships

  equation289

Then, another way to write the radial distance r is

  equation298

To find the eccentric anomaly E, one must solve the transcendental equation

  equation301

where tex2html_wrap_inline1704 is the coordinate time of perigee passage.

In Newtonian mechanics, the gravitational field is a conservative field and total energy is conserved. Using the above equations for the Keplerian orbit, one can show that the total energy per unit mass of the satellite is

  equation307

If I use Eq. (33Popup Equation) for tex2html_wrap_inline1706 in Eq. (28Popup Equation), then I get the following expression for the elapsed coordinate time on the satellite clock:

  equation315

The first two constant rate correction terms in Eq. (34Popup Equation) have the values:

  equation325

The negative sign in this result means that the standard clock in orbit is beating too fast, primarily because its frequency is gravitationally blueshifted. In order for the satellite clock to appear to an observer on the geoid to beat at the chosen frequency of 10.23 MHz, the satellite clocks are adjusted lower in frequency so that the proper frequency is:

  equation333

This adjustment is accomplished on the ground before the clock is placed in orbit.

  

Click on thumbnail to view image

Figure 2: Net fractional frequency shift of a clock in a circular orbit.

Figure  2 shows the net fractional frequency offset of an atomic clock in a circular orbit, which is essentially the left side of Eq. (35Popup Equation) plotted as a function of orbit radius a, with a change of sign. Five sources of relativistic effects contribute in Figure  2 . The effects are emphasized for several different orbit radii of particular interest. For a low earth orbiter such as the Space Shuttle, the velocity is so great that slowing due to time dilation is the dominant effect, while for a GPS satellite clock, the gravitational blueshift is greater. The effects cancel at tex2html_wrap_inline1710 . The Global Navigation Satellite System Galileo, which is currently being designed under the auspices of the European Space Agency, will have orbital radii of approximately 30,000 km.

There is an interesting story about this frequency offset. At the time of launch of the NTS-2 satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was recognized that orbiting clocks would require a relativistic correction, but there was uncertainty as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic effects were truths that would need to be incorporated [5]! A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by general relativity, then the synthesizer could be turned on, bringing the clock to the coordinate rate necessary for operation. After the Cesium clock was turned on in NTS-2, it was operated for about 20 days to measure its clock rate before turning on the synthesizer [11]. The frequency measured during that interval was +442.5 parts in tex2html_wrap_inline1712 compared to clocks on the ground, while general relativity predicted +446.5 parts in tex2html_wrap_inline1712 . The difference was well within the accuracy capabilities of the orbiting clock. This then gave about a 1% verification of the combined second-order Doppler and gravitational frequency shift effects for a clock at 4.2 earth radii.

Additional small frequency offsets can arise from clock drift, environmental changes, and other unavoidable effects such as the inability to launch the satellite into an orbit with precisely the desired semimajor axis. The navigation message provides satellite clock frequency corrections for users so that in effect, the clock frequencies remain as close as possible to the frequency of the U.S.\ Naval Observatory's reference clock ensemble. Because of such effects, it would now be difficult to use GPS clocks to measure relativistic frequency shifts.

When GPS satellites were first deployed, the specified factory frequency offset was slightly in error because the important contribution from earth's centripetal potential (see Eq. (18)) had been inadvertently omitted at one stage of the evaluation. Although GPS managers were made aware of this error in the early 1980s, eight years passed before system specifications were changed to reflect the correct calculation [2Jump To The Next Citation Point In The Article]. As understanding of the numerous sources of error in the GPS slowly improved, it eventually made sense to incorporate the correct relativistic calculation.

The eccentricity correction.   The last term in Eq. (34Popup Equation) may be integrated exactly by using the following expression for the rate of change of eccentric anomaly with time, which follows by differentiating Eq. (32Popup Equation):

equation358

Also, since a relativistic correction is being computed, tex2html_wrap_inline1716, so

  eqnarray362

The constant of integration in Eq. (38Popup Equation) can be dropped since this term is lumped with other clock offset effects in the Kalman filter computation of the clock correction model. The net correction for clock offset due to relativistic effects that vary in time is

  equation378

This correction must be made by the receiver; it is a correction to the coordinate time as transmitted by the satellite. For a satellite of eccentricity e = 0.01, the maximum size of this term is about 23 ns. The correction is needed because of a combination of effects on the satellite clock due to gravitational frequency shift and second-order Doppler shift, which vary due to orbit eccentricity.

Eq. (39Popup Equation) can be expressed without approximation in the alternative form

  equation387

where tex2html_wrap_inline1524 and tex2html_wrap_inline1722 are the position and velocity of the satellite at the instant of transmission. This may be proved using the expressions (30Popup Equation, 31Popup Equation, 32Popup Equation) for the Keplerian orbits of the satellites. This latter form is usually used in implementations of the receiver software.

It is not at all necessary, in a navigation satellite system, that the eccentricity correction be applied by the receiver. It appears that the clocks in the GLONASS satellite system do have this correction applied before broadcast. In fact historically, this was dictated in the GPS by the small amount of computing power available in the early GPS satellite vehicles. It would actually make more sense to incorporate this correction into the time broadcast by the satellites; then the broadcast time events would be much closer to coordinate time - that is, GPS system time. It may now be too late to reverse this decision because of the investment that many dozens of receiver manufacturers have in their products. However, it does mean that receivers are supposed to incorporate the relativity correction; therefore, if appropriate data can be obtained in raw form from a receiver one can measure this effect. Such measurements are discussed next.



6 TOPEX/POSEIDON Relativity ExperimentRelativity in the Global Positioning 4 The Realization of Coordinate

image Relativity in the Global Positioning System
Neil Ashby
http://www.livingreviews.org/lrr-2003-1
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de