Five distinct relativistic effects, discussed in Section 5, are incorporated into the System Specification Document, ICD-GPS-200 [2]. 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:
where
v
is the satellite speed in a local ECI reference frame,
is the product of the Newtonian gravitational constant
G
and earth's mass
M,
c
is the defined speed of light, and
is the effective gravitational potential on the earth's rotating
geoid. The term
includes contributions from both monopole and quadrupole moments
of earth's mass distribution, and the effective centripetal
potential in an earth-fixed reference frame such as the
WGS-84(837) frame, due to earth's rotation. The value for
is given in Eq. (18
), and depends on earth's equatorial radius
, earth's quadrupole moment coefficient
, and earth's angular rotational speed
.
If the GPS satellite orbit can be approximated by a Keplerian
orbit of semi-major axis
a, then at an instant when the distance of the clock from earth's
center of mass is
r, this leads to the following expression for the fraction
frequency shift of Eq. (53):
Eq. (54) is derived by making use of the conservation of total energy
(per unit mass) of the satellite, Eq. (33
), which leads to an expression for
in terms of
and
that can be substituted into Eq. (53
). The first two terms in Eq. (54
) give rise to the ``factory frequency offset'', which is applied
to GPS clocks before launch in order to make them beat at a rate
equal to that of reference clocks on earth's surface. The last
term in Eq. (54
) is very small when the orbit eccentricity
e
is small; when integrated over time these terms give rise to the
so-called ``
'' effect or ``eccentricity effect''. In most of the following
discussion we shall assume that eccentricity is very small.
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
The factor 3/2 in this expression arises from the combined effect of second-order Doppler and gravitational frequency shifts. If the semi-major axis increases, the satellite will be higher in earth's gravitational potential and will be gravitationally blue-shifted more, while at the same time the satellite velocity will be reduced, reducing the size of the second-order Doppler shift (which is generally a red shift). The net effect would make a positive contribution to the fractional frequency shift.
Although it has long been known that orbit adjustments are
associated with satellite clock frequency shifts, nothing has
been documented and up until recently no reliable measurements of
such shifts have 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. Recently, Marvin Epstein, Joseph Fine, and Eric
Stoll [12] of ITT have 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.
[12] 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:
Since the semi-major axis decreased, the frequency shift
should be negative. The prediction they made for the frequency
shift, which was based on Eq. (53), was then
which is to be compared with the measured value, Eq. (56). This is fairly compelling evidence that the observed frequency
shift is indeed a relativistic effect.
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 [13]. 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 1/
e
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 (x, y, z) in equatorial ECI coordinates in the model under consideration here is
Since the force is conservative in this model (solar radiation
pressure, thrust,
etc.
are not considered), the kinetic plus potential energy is
conserved. Let
be the energy per unit mass of an orbiting mass point. Then
where
V
'(x,
y,
z) is the perturbing potential due to the earth's quadrupole
potential. It is shown in textbooks [13] that, with the help of Lagrange's planetary perturbation
theory, the conservation of energy condition can be put in the
form
where a is the perturbed (osculating) semi-major axis. In other words, for the perturbed orbit,
On the other hand, the net fractional frequency shift relative to a clock at rest at infinity is determined by the second-order Doppler shift (a red shift) and a gravitational redshift. The total relativistic fractional frequency shift is
The conservation of energy condition can be used to express
the second-order Doppler shift in terms of the potential. Since
in this paper we are interested in fractional frequency changes
caused by changing the orbit, it will make no difference if the
calculations use a clock at rest at infinity as a reference
rather than a clock at rest on earth's surface. The reference
potential cancels out to the required order of accuracy.
Therefore, from perturbation theory we need expressions for the
square of the velocity, for the radius
r, and for the perturbing potential. We now proceed to derive
these expressions. We refer to the literature [13] 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
e
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 E is to be calculated by solving the equation
where
M
is the ``mean anomaly'' and
is the time of passage past perigee, and
Then, the perturbed radial distance r and true anomaly f of the satellite are obtained from
The observable x, y, z -coordinates of the satellite are then calculated from the following equations:
where
is the angle of the ascending line of nodes,
i
is the inclination, and
is the altitude of perigee. By differentiation with respect to
time, or by using the conservation of energy equation, one
obtains the following expression for the square of the
velocity:
In these expressions
and
are observable quantities. The combination
, where E is the eccentric anomaly, occurs in both of these
expressions. To derive expressions for
and
in the perturbed orbits, expressions for the perturbed elements
a,
e,
E
are to be substituted into the right-hand sides of the Keplerian
equations for
E,
r, and
. Therefore, we need the combination
in the limit of small eccentricity.
Perturbed eccentricity. To leading order, from the literature [13] we have for the perturbed eccentricity the following expression:
where
is a constant of integration.
Perturbed eccentric anomaly. The eccentric anomaly is calculated from the equation
with perturbed values for
M
and
e
. Expanding to first order in
e
gives the following expression for
:
and multiplying by e yields
We shall neglect higher order terms in e . The perturbed expression for mean anomaly M can be written as
where we indicate explicitly the terms in
; that is, the quantity
contains all terms which do not blow up as
, and
contains all the other terms. The perturbations of
M
are known exactly but we shall only need the leading order
terms, which are
and so for very small eccentricity,
Then after accounting for contributions from the perturbed
eccentricity and the perturbed mean anomaly, after a few lines of
algebra we obtain the following for
:
where the first term is the unperturbed part. The perturbation is a constant, plus a term with twice the orbital period.
Perturbation in semi-major axis. From the literature, the leading terms in the perturbation of the semi-major axis are
where
is a constant of integration. The amplitude of the periodic term
is about 1658 meters.
Perturbation in radius. We are now in position to compute the perturbation in the radius. From the expression for r, after combining terms we have
The amplitude of the periodic part of the perturbation in the observable radial distance is only 276 meters.
Perturbation in the velocity squared.
The above results, after substituting into Eq. (70), yield the expression
Perturbation in
.
The above expression for the perturbed
r
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
r
and
z
into
V
'(x,
y,
z), 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
This verifies that the perturbation theory gives a constant
energy. The extra term in the above equation, with
in it, can be neglected. This is because the nominal inclination
of GPS orbits is such that the factor
is essentially zero. The near vanishing of this factor is pure
coincidence in the GPS. There was no intent, in the original GPS
design, that quadrupole effects would be simpler if the orbital
inclination were close to
. 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 first term, when combined with the reference potential at earth's geoid, gives rise to the ``factory frequency offset''. The seond term gives rise to the eccentricity effect. The third term can be neglected, as pointed out above. The last term has an amplitude
which may be large enough to consider when calculating frequency shifts produced by orbit changes. Therefore, this contribution may have to be considered in the future in the determination of the semi-major axis, but for now we neglect it.
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 be
The last periodic term in Eq. (85) is of a form similar to that which gives rise to the
eccentricity correction, which is applied by GPS receivers.
Considering only the last periodic term, the additional time
elapsed on the orbiting clock will be given by
where to a sufficient approximation we have replaced the
quantity
f
in the integrand by
;
n
is the approximate mean motion of GPS satellites. Integrating
and dropping the constant of integration (assuming as usual that
such constant time offsets are lumped with other contributions)
gives the periodic relativistic effect on the elapsed time of the
SV clock due to earth's quadrupole moment:
The correction that should be applied by the receiver is the negative of this expression,
The phase of this correction is zero when the satellite passes
through earth's equatorial plane going northwards. If not
accounted for, this effect on the SV clock time would give rise
to a peak-to-peak periodic navigational error in position of
approximately
.
These effects were considered by Ashby and Spilker [9], 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:
Therefore, the fractional frequency change produced by the orbit change of July 25 is calculated to be
which agrees with the measured value to within about 3.3%. The
agreement is slightly better than that obtained in [12], perhaps because they did not consider contributions to the
energy from the quadrupole moment term.
A similar calculation shows that the fractional frequency shift of SV43 on October 10, 2001 should have been
This shift has not yet been measured accurately.
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
Using Eq. (55) yields the following prediction for the fractional frequency
change of SV54 on March 9, 2001:
The quoted uncertainty is due to the combined uncertainties from the determination of the energy per unit mass before and after the orbit change. These uncertainties are due to neglecting tidal forces of the sun and moon, radiation pressure, and other non-gravitational forces.
Summary. We note that the values of semi-major axis reported by Epstein et al. [12] 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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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 |