I found no measurements there.
Words only.
You have words only too.
You keep repeating this same silly statement over and over. Since you are either too lazy to follow the link given OR too dense to even try, I'll post a part of the info here that most certainly DOES include the measurements!!!!! If you still deny that there aren't any measurements present then you have shown yourself to be one of the greatest fools in the entire world!!
Here:
"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. (28). The term in Eq. (28) includes the scale correction needed in order to use clocks at rest on the earth’s surface as references. The quadrupole contributes to in the term in Eq. (28); there it contributes a fractional rate correction of –3.76 × 10–13. This effect must be accounted for in the GPS. Also, 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 1014, 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. (28). Since the quadrupole (and higher multipole) parts of the earth’s potential are neglected, in Eq. (28) the potential is . 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 and eccentricity by . Then the solution of the orbital equations is as follows [13]: The distance from the center of the earth to the satellite in ECI coordinates is
The angle , 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 called the eccentric anomaly, according to the relationships
Then, another way to write the radial distance is
To find the eccentric anomaly , one must solve the transcendental equation
where 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
If I use Eq. (33) for in Eq. (28), then I get the following expression for the elapsed coordinate time on the satellite clock:
The first two constant rate correction terms in Eq. (34) have the values:
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:
This adjustment is accomplished on the ground before the clock is placed in orbit.
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. (35) plotted as a function of orbit radius , 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 . 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 1012 compared to clocks on the ground, while general relativity predicted +446.5 parts in 1012. 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 [2]. As understanding of the numerous sources of error in the GPS slowly improved, it eventually made sense to incorporate the correct relativistic calculation. It has become common practice not to apply such offsets to Rubidium clocks as these are subject to unpredictable frequency jumps during launch. Instead, after such clocks are placed in orbit their frequencies are measured and the actual frequency corrections needed are incorporated in the clock correction polynomial that accompanies the navigation message.Update
The eccentricity correction. The last term in Eq. (34) may be integrated exactly by using the following expression for the rate of change of eccentric anomaly with time, which follows by differentiating Eq. (32):
Also, since a relativistic correction is being computed, , so
The constant of integration in Eq. (38) 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
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 , 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. (39) can be expressed without approximation in the alternative form
where and are the position and velocity of the satellite at the instant of transmission. This may be proved using the expressions (30, 31, 32) 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.
A report distributed by the Aerospace Corporation [14] has claimed that the correction expressed in Eqs. (38) and (39) would not be valid for a highly dynamic receiver – e.g., one in a highly eccentric orbit. This is a conceptual error, emanating from an apparently official source, which would have serious consequences. The GPS modernization program involves significant redesign and remanufacturing of the Block IIF satellites, as well as a new generation of satellites that are now being deployed – the Block IIR replenishment satellites. These satellites are capable of autonomous operation, that is, they can be operated independently of the ground-based control segment for up to 180 days. They are to accomplish this by having receivers on board that determine their own position and time by listening to the other satellites that are in view. If the conceptual basis for accounting for relativity in the GPS, as it has been explained above, were invalid, the costs of opening up these satellites and reprogramming them would be astronomical.
There has been therefore considerable controversy about this issue. As a consequence, it was proposed by William Feess of the Aerospace Corporation that a measurement of this effect be made using the receiver on board the TOPEX satellite. The TOPEX satellite carries an advanced, six-channel GPS receiver. With six data channels available, five of the channels can be used to determine the bias on the local oscillator of the TOPEX receiver with some redundancy, and data from the sixth channel can be used to measure the eccentricity effect on the sixth SV clock. Here I present some preliminary results of these measurements, which are to my knowledge the only explicit measurements of the periodic part of the combined relativistic effects of time dilation and gravitational frequency shift on an orbiting receiver.
A brief description of the pseudorange measurement made by a receiver is needed here before explaining the TOPEX data. Many receivers work by generating a replica of the coded signal emanating from the transmitter. This replica, which is driven through a feedback shift register at a rate matching the Doppler-shifted incoming signal, is correlated with the incoming signal. The transmitted coordinate time can be identified in terms of a particular phase reversal at a particular point within the code train of the signal. When the correlator in the receiver is locked onto the incoming signal, the time delay between the transmission event and the arrival time, as measured on the local clock, can be measured at any chosen instant.
Let the time as transmitted from the satellite be denoted by . After correcting for the eccentricity effect, the GPS time of transmission would be . Because of SA (which was in effect for the data that were chosen), frequency offsets and frequency drifts, the satellite clock may have an additional error so that the true GPS transmission time is .
Now the local clock, which is usually a free-running oscillator subject to various noise and drift processes, can be in error by a large amount. Let the measured reception time be and the true GPS time of reception be . The possible existence of this local clock bias is the reason why measurements from four satellites are needed for navigation, as from four measurements the three components of the receiver’s position vector, and the local clock bias, can be determined. The raw difference between the time of reception of the time tag from the satellite, and the time of transmission, multiplied by , is an estimate of the geometric range between satellite and receiver called the pseudorange [22]:
On the other hand the true range between satellite and receiver is
Combining Eqs. (41)and (42) yields the measurement equation for this experiment:
The purpose of the TOPEX satellite is to measure the height of the sea. This satellite has a six-channel receiver on board with a very good quartz oscillator to provide the time reference. A radar altimeter measures the distance of the satellite from the surface of the sea, but such measurements play no role in the present experiment. The TOPEX satellite has orbit radius 7,714 km, an orbital period of about 6745 seconds, and an orbital inclination of 66.06° to earth’s equatorial plane. Except for perturbations due to earth’s quadrupole moment, the orbit is very nearly circular, with eccentricity being only 0.000057. The TOPEX satellite is almost ideal for analysis of this relativity effect. The trajectories of the TOPEX and GPS satellites were determined independently of the on-board clocks, by means of Doppler tracking from 100 stations maintained by the Jet Propulsion Laboratory (JPL).
The receiver is a dual frequency C/A- and P-code receiver from which both code data and carrier phase data were obtained. The dual-frequency measurements enabled us to correct the propagation delay times for electron content in the ionosphere. Close cooperation was given by JPL and by William Feess in providing the dual-frequency measurements, which are ordinarily denied to civilian users, and in removing the effect of SA at time points separated by 300 seconds during the course of the experiment.
The following data were provided through the courtesy of Yoaz Bar-Sever of JPL for October 22–23, 1995:
ECI center-of-mass position and velocity vectors for 25 satellites, in the J2000 Coordinate system with times in UTC. Data rate is every 15 minutes; accuracy quoted is 10 cm radial, 30 cm horizontal.
ECI position and velocity vectors for the TOPEX antenna phase center. Data rate is every minute in UTC; accuracy quoted is 3 cm radial and 10 cm horizontal.
GPS satellite clock data for 25 satellites based on ground system observations. Data rate is every 5 minutes, in GPS time; accuracy ranges between 5 and 10 cm.
TOPEX dual frequency GPS receiver measurements of pseudorange and carrier phase for 25 satellites, a maximum of six at any one time. The data rate is every 10 seconds, in GPS time.
During this part of 1995, GPS time was ahead of UTC by 10 seconds. GPS cannot tolerate leap seconds so whenever a leap second is inserted in UTC, UTC falls farther behind GPS time. This required high-order interpolation on the orbit files to obtain positions and velocities at times corresponding to times given, every 300 seconds, in the GPS clock data files. When this was done independently by William Feess and myself we agreed typically to within a millimeter in satellite positions.
The L1 and L2 carrier phase data was first corrected for ionospheric delay. Then the corrected carrier phase data was used to smooth the pseudorange data by weighted averaging. SA was compensated in the clock data by courtesy of William Feess. Basically, the effect of SA is contained in both the clock data and in the pseudorange data and can be eliminated by appropriate subtraction. Corrections for the offset of the GPS SV antenna phase centers from the SV centers of mass were also incorporated.
The determination of the TOPEX clock bias is obtained by rearranging Eq. (43):
Generally, at each time point during the experiment, observations were obtained from six (sometimes five) satellites. The geometric range, the first term in Eq. (44), was determined by JPL from independent Doppler tracking of both the GPS constellation and the TOPEX satellite. The pseudorange was directly measured by the receiver, and clock models provided the determination of the clock biases in the satellites. The relativity correction for each satellite can be calculated directly from the given GPS satellite orbits. Because the receiver is a six-channel receiver, there is sufficient redundancy in the measurements to obtain good estimates of the TOPEX clock bias and the rms error in this bias due to measurement noise. The resulting clock bias is plotted in Figure 3.
Figure 3: TOPEX clock bias in meters determined from 1,571 observations.
Figure 4: Rms deviation from mean of TOPEX clock bias determinations.
The rms deviation from the mean of the TOPEX clock biases is plotted in Figure 4 as a function of time. The average rms error is 29 cm, corresponding to about one ns of propagation delay. Much of this variation can be attributed to multipath effects.
Figure 3 shows an overall frequency drift, accompanied by frequency adjustments and a large periodic variation with period equal to the orbital period. Figure 3 gives our best estimate of the TOPEX clock bias. This may now be used to measure the eccentricity effects by rearranging Eq. (43):
Strictly speaking, in finding the eccentricity effect this way for a particular satellite, one should not include data from that satellite in the determination of the clock bias. One can show, however, that the penalty for this is simply to increase the rms error by a factor of 6/5, to 35 cm. Figure 4 plots the rms errors in the TOPEX clock bias determination of Figure 3. Figure 5 shows the measured eccentricity effect for SV nr. 13, which has the largest eccentricity of the satellites that were tracked, . The solid curve in Figure 5 is the theoretically predicted effect, from Eq. (39). While the agreement is fairly good, one can see some evidence of systematic bias during particular passes, where the rms error (plotted as vertical lines on the measured dots) is significantly smaller than the discrepancies between theory and experiment. For this particular satellite, the rms deviation between theory and experiment is 22 cm, which is about 2.2% of the maximum magnitude of the effect, 10.2 m. Update
Figure 5: Comparison of predicted and measured eccentricity effect for SV nr. 13.
Figure 6: Generic eccentricity effect for five satellites.
Similar plots were obtained for 25 GPS satellites that were tracked during this experiment. Rather than show them one by one, it is interesting to plot them on the same graph by dividing the calculated and measured values by eccentricity , while translating the time origin so that in each case time is measured from the instant of perigee passage. We plot the effects, not the corrections. In this way, Figure 6 combines the eccentricity effects for the five satellites with the largest eccentricities. These are SV’s nr. 13, 21, 27, 23, and 26. In Figure 6 the systematic deviations between theory and experiment tend to occur for one satellite during a pass; this “pass bias” might be removable if we understood better what the cause of it is. As it stands, the agreement between theory and experiment is within about 2.5%."