Improvements in GPS motivate attention to other small relativistic effects that have previously been too small to be explicitly considered. For SV clocks, these include frequency changes due to orbit adjustments, and effects due to earth’s oblateness. For example, between July 25 and October 10, 2000, SV43 occupied a transfer orbit while it was moved from slot 5 to slot 3 in orbit plane F. I will show here that the fractional frequency change associated with a change da in the semi-major axis a (in meters) can be estimated as 9.429 × 10–18 da. In the case of SV43, this yields a prediction of –1.77 × 10–13 for the fractional frequency change of the SV43 clock which occurred July 25, 2000. This relativistic effect was measured very carefully . Another orbit adjustment on October 10, 2000 should have resulted in another fractional frequency change of +1.75 × 10–13, which was not measured carefully. Also, earth’s oblateness causes a periodic fractional frequency shift with period of almost 6 hours and amplitude 0.695 × 10–14. This means that quadrupole effects on SV clock frequencies are of the same general order of magnitude as the frequency breaks induced by orbit changes. Thus, some approximate expressions for the frequency effects on SV clock frequencies due to earth’s oblateness are needed. These effects will be discussed with the help of Lagrange’s planetary perturbation equations.
Five distinct relativistic effects, discussed in Section 5, are incorporated into the System Specification Document, ICD-GPS-200 . These are:
The combination of second-order Doppler and gravitational frequency shifts given in Eq. (27) for a clock in a GPS satellite leads directly to the following expression for the fractional frequency shift of a satellite clock relative to a reference clock fixed on earth’s geoid:Update
If the GPS satellite orbit can be approximated by a Keplerian orbit of semi-major axis , then at an instant when the distance of the clock from earth’s center of mass is , this leads to the following expression for the fraction frequency shift of Eq. (53):
Clearly, from Eq. (54), if the semi-major axis should change by an amount due to an orbit adjustment, the satellite clock will experience a fractional frequency change
Although it has long been known that orbit adjustments are associated with satellite clock frequency shifts, nothing has been documented and up until 2000 no reliable measurements of such shifts had been made. On July 25, 2000, a trajectory change was applied to SV43 to shift the satellite from slot F5 to slot F3. A drift orbit extending from July 25, 2000 to October 10, 2000 was used to accomplish this move. A “frequency break” was observed but the cause of this frequency jump was not initially understood. Marvin Epstein, Joseph Fine, and Eric Stoll  of ITT evaluated the frequency shift of SV43 arising from this trajectory change. They reported that associated with the thruster firings on July 25, 2000 there was a frequency shift of the Rubidium clock on board SV43 of amount
Epstein et al.  suggested that the above frequency shift was relativistic in origin, and used precise ephemerides obtained from the National Imagery and Mapping Agency to estimate the frequency shift arising from second-order Doppler and gravitational potential differences. They calculated separately the second-order Doppler and gravitational frequency shifts due to the orbit change. The NIMA precise ephemerides are expressed in the WGS-84 coordinate frame, which is earth-fixed. If used without removing the underlying earth rotation, the velocity would be erroneous. They therefore transformed the NIMA precise ephemerides to an earth-centered inertial frame by accounting for a (uniform) earth rotation rate.
The semi-major axes before and after the orbit change were calculated by taking the average of the maximum and minimum radial distances. Speeds were calculated using a Keplerian orbit model. They arrived at the following numerical values for semi-major axis and velocity:
Lagrange perturbation theory. Perturbations of GPS orbits due to earth’s quadrupole mass distribution are a significant fraction of the change in semi-major axis associated with the orbit change discussed above. This raises the question whether it is sufficiently accurate to use a Keplerian orbit to describe GPS satellite orbits, and estimate the semi-major axis change as though the orbit were Keplerian. In this section, we estimate the effect of earth’s quadrupole moment on the orbital elements of a nominally circular orbit and thence on the change in frequency induced by an orbit change. Previously, such an effect on the SV clocks has been neglected, and indeed it does turn out to be small. However, the effect may be worth considering as GPS clock performance continues to improve.
To see how large such quadrupole effects may be, we use exact calculations for the perturbations of the Keplerian orbital elements available in the literature . For the semi-major axis, if the eccentricity is very small, the dominant contribution has a period twice the orbital period and has amplitude . WGS-84 (837) values for the following additional constants are used in this section: ; ; , where and are earth’s equatorial radius and SV orbit semi-major axis, and is earth’s rotational angular velocity.
The oscillation in the semi-major axis would significantly affect calculations of the semi-major axis at any particular time. This suggests that Eq. (33) needs to be reexamined in light of the periodic perturbations on the semi-major axis. Therefore, in this section we develop an approximate description of a satellite orbit of small eccentricity, taking into account earth’s quadrupole moment to first order. Terms of order will be neglected. This problem is non-trivial because the perturbations themselves (see, for example, the equations for mean anomaly and altitude of perigee) have factors which blow up as the eccentricity approaches zero. This problem is a mathematical one, not a physical one. It simply means that the observable quantities – such as coordinates and velocities – need to be calculated in such a way that finite values are obtained. Orbital elements that blow up are unobservable.
Conservation of energy. The gravitational potential of a satellite at position in equatorial ECI coordinates in the model under consideration here isetc. are not considered), the kinetic plus potential energy is conserved. Let be the energy per unit mass of an orbiting mass point. Then  that, with the help of Lagrange’s planetary perturbation theory, the conservation of energy condition can be put in the form  for the perturbed osculating elements. These are exactly known, to all orders in the eccentricity, and to first order in . We shall need only the leading terms in eccentricity for each element.
Perturbation equations. First we recall some facts about an unperturbed Keplerian orbit, which have already been introduced (see Section 5). The eccentric anomaly is to be calculated by solving the equation
Perturbed eccentricity. To leading order, from the literature  we have for the perturbed eccentricity the following expression:
Perturbed eccentric anomaly. The eccentric anomaly is calculated from the equation
Perturbation in semi-major axis. From the literature, the leading terms in the perturbation of the semi-major axis are
Perturbation in radius. We are now in position to compute the perturbation in the radius. From the expression for , after combining terms we have
Perturbation in the velocity squared. The above results, after substituting into Eq. (70), yield the expression
Perturbation in . The above expression for the perturbed yields the following for the monopole contribution to the gravitational potential:
Evaluation of the perturbing potential. Since the perturbing potential contains the small factor , to leading order we may substitute unperturbed values for and into , which yields the expression
Conservation of energy. It is now very easy to check conservation of energy. Adding kinetic energy per unit mass to two contributions to the potential energy gives°. However, because this term is negligible, numerical calculations of the total energy per unit mass provide a means of evaluating the quantity .
Calculation of fractional frequency shift. The fractional frequency shift calculation is very similar to the calculation of the energy, except that the second-order Doppler term contributes with a negative sign. The result is
The result suggests the following method of computing the fractional frequency shift: Averaging the shift over one orbit, the periodic term will average down to a negligible value. The third term is negligible. So if one has a good estimate for the nominal semi-major axis parameter, the term gives the average fractional frequency shift. On the other hand, the average energy per unit mass is given by . Therefore, the precise ephemerides, specified in an ECI frame, can be used to compute the average value for ; then the average fractional frequency shift will benegative of this expression,
These effects were considered by Ashby and Spilker , pp. 685–686, but in that work the effect of earth’s quadrupole moment on the term was not considered; the present calculations supercede that work.
Numerical calculations. Precise ephemerides were obtained for SV43 from the web site ftp://sideshow.jpl.nasa.gov/pub/gipsy_products/2000/orbits at the Jet Propulsion Laboratory. These are expressed in the J2000 ECI frame. Computer code was written to compute the average value of for one day and thence the fractional frequency shift relative to infinity before and after each orbit change. The following results were obtained:
A similar calculation shows that the fractional frequency shift of SV43 on October 10, 2001 should have been
On March 9, 2001, SV54’s orbit was changed by firing the thruster rockets. Using the above procedures, I can calculate the fractional frequency change produced in the onboard clocks. The result is
Summary. We note that the values of semi-major axis reported by Epstein et al.  differ from the values obtained by averaging as outlined above, by 200–300 m. This difference arises because of the different methods of calculation. In the present calculation, an attempt was made to account for the effect of earth’s quadrupole moment on the Keplerian orbit. It was not necessary to compute the orbit eccentricity. Agreement with measurement of the fractional frequency shift was only a few percent better than that obtained by differencing the maximum and minimum radii. This approximate treatment of the orbit makes no attempt to consider perturbations that are non-gravitational in nature, e.g., solar radiation pressure. The work was an investigation of the approximate effect of earth’s quadrupole moment on the GPS satellite orbits, for the purpose of (possibly) accurate calculations of the fractional frequency shifts that result from orbit changes.
As a general conclusion, the fractional frequency shift can be estimated to very good accuracy from the expression for the “factory frequency offset”.
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