THE PERTURBATION OF THE ORBITAL ELEMENTS OF GPS
SATELLITES THROUGH DIRECT RADIATION PRESSURE
AND Y BIAS
Markus Rothacher Gerhard Beutler Leo sMervart
Astronomical Institute University of Berne
CH Bern Switzerland
ABSTRACT
A direct radiation pressure p and a constant p so called y bias are estimated for each
0 2
GPS satellite and each arc of one to three days on top of the a priori radiation pressure
mo del de ned by the IERS Standards in the daily analyses p erformed at the CODE Analysis
Center of the IGS
In this article we are analyzing the in uence of the two estimated parameters on the orbital
elements using the metho ds of sp ecial p erturbations We develop approximated expressions
for the p erturbations in the Keplerian elements which allow us to study the p erturbations
analytically We are in particular interested in the correlations of these parameters with the
length of day UT UTC drift estimates and the so called pseudo stochastic parameters
which are also solved for routinely at CODE Using the IGS data stemming from a time
interval of days in we show that the theoretical results are con rmed in practice
INTRODUCTION
CODE Center for Orbit Determination in Europ e is one of at present seven Analysis
Centers of the IGS the International GPS Service for Geodynamics CODE is formed as
a joint venture of the Astronomical Institute of the University of Bern AIUB the Swiss
Federal O ce of Topography L T the German Institute for Applied Geo desy IfAG
and the French Institut G eographique National IGN CODE is lo cated at the AIUB in
Bern
Since June ephemerides for all GPS satellites are available and daily values for the
earth orientation parameters x and y of the p ole p osition and drift values for UT UTC
are solved for and made available to the scienti c community by CODE For test purp oses
the drifts and in the nutation corrections are determined for test purp oses These
values are not yet made available to the community
Other parameters like station co ordinates and their velocities are estimated by combining
the daily solutions So far they were pro duced for the annual solutions of the IERS the
International Earth Rotation Service Paper presented at the IGS Workshop Potsdam Germany May
The CODE pro ducts are analysed and compared to other solutions stemming from GPS
or from other space techniques like VLBI and SLR Whereas in general the CODE results
compare well to all the other results there is a slight problem in the UT UTC drift rates
as determined by CODE This problem is not obvious if we just compare our length of day
estimates to those established by VLBI there clearly all the signatures of a physical nature
are present Figure shows our length of day estimates after removal of the tidal terms up
to days
Figure CODE length of day estimates after removal of tidal terms
That there is a small bias with resp ect to the VLBI b ecomes obvious if we sum up our
daily UT UTC drifts and subtract the corresp onding VLBI UT UTC series from our
series Figure shows the result There is a net drift b etween the two series There also
seems to b e a net change of the drift around May
Figure CODE UT UTC series minus corresp onding VLBI series actually the IERS
Bulletin A series
It is well known that satellite geo desy not having direct access to direction observations
in the celestial reference frame is not able to determine UT UTC itself b ecause of the
correlation with the right ascensions of the ascending no des of the GPS satellites which
have to b e solved for in each arc
Could we assume that the force eld acting on GPS satellites is completely known we
would b e in a p erfect p osition to solve for the UT UTC drift however This statement
is almost true in the case of GPS satellites The earth s gravity eld may b e assumed
known for these high orbiting satellites third b o dy p erturbations and tidal e ects do not
p ose problems On the other hand we have to solve for radiation pressure parameters for
each arc and each satellite Should these parameters generate among other e ects a net
rotation of the orbital system a correlation with the UT UTC drift estimates would b e
p ossible This asp ect is studied on a theoretical basis in the next section empirical evidence
is presented in the subsequent section
PERTURBATIONS DUE TO RADIATION PRESSURE ORDERS OF
MAGNITUDE
In this section we study the in uence of the radiation pressure mo del used at CODE on
the estimated orbits using the metho d of sp ecial p erturbations For that purp ose we rst
have to develop analytical expressions for the p erturbing forces then we have to present
simpli ed p erturbation equations We assume in particular that GPS orbits are almost
circular Finally we will develop and discuss the e ects and p ossible correlations of radiation
pressure estimates with earth rotation parameters
Analytical Expressions for the Radiation Pressure
Figure shows the pro jection on a unit sphere of the geo centric p ositions of the satellite
and the sun at a particular instant of time t For the description of the p ositions of satellite
and sun we introduce two co ordinate systems one which is describ ed by the axes x x
1 2
and x in Figure and the other which is describ ed by the axes y y and y The latter
3 1 2 3
system is rotating with the satellite it may b e generated from the rst co ordinate system
by a rotation around the x y axis with the argument of latitude u of the satellite as the
3 3 rotation angle
x = y 3 3 y 2 x 2
y1
u β 0 Ω u 0 i
x1
Figure Positions of satellite and sun u and u are the arguments of latitude of satellite
0
and sun is the elevation of the sun ab ove the orbital plane is the right
0
ascension of the ascending no de of the satellite s orbital plane
The co ordinates r of the satellite p osition r and the co ordinates e of the unit vector e
x s x
p ointing from the satellite to the sun may b e expressed as follows in the co ordinate system
fx x x g
1 2 3
cos u
B C
r r sin u
A
x
cos u cos
0 0
B C
e sin u cos
A
s x 0 0
sin
0
where we assume that the direction satel lite sun may b e approximated by the direction
center of earth sun
The p erturbation equations given in the next section ask for the radial the tangential
and the out of plane comp onents R S and W of the p erturbing forces We thus have to
express the two p osition vectors in the co ordinate system fy y y g
1 2 3
B C
r R u r r
A
y 3 x
cos u u cos
0 0
B C
e R u e sin u u cos
A
s y 3 s x 0 0
sin
0
where R u stands for a rotation matrix describing a particular rotation around axis and
3
angle u
At CODE the acceleration a due to the solar radiation pressure is mo deled as Beutler
dr p
et al
a a p e p e
dr p Rock 0 s 2 y
where
a is the standard mo del Ro ck for Blo ck I the standard mo del Ro ck for Blo ck II
Rock
satellites Fliegel et al
e is the unit vector satel lite sun in our approximation the unit vector center of earth
s
sun
e is the unit vector p ointing along the spacecraft s solar panels axis Fliegel et al
y
e r
s
e
y
j e r j
s
p and p are the direct radiation pressure parameters estimated for each satellite and each
0 2
arc
It should b e p ointed out that in a rst approximation the vector a is parallel to the
Rock
vector p e We therefore do not analyse the term stemming from the Ro ck mo dels
0 s
separately By forming the vector pro duct in the co ordinate system fy y y g we obtain
1 2 3
a a a a
dr p y Rock y 0 y 2 y
cos u u cos
0 0
B C
a p sin u u cos
A
Rock y 0 0 0
sin
0
p
B C
2
q
sin
A
0
2 2
2
sin cos sin u u
0 0 0
cos sin u u
0 0
It is of interest to write down the equation for the cases corresp onding to the
0
2
case when the sun is in the zenith of the orbital plane and corresp onding to the
0
case where the the sun lies in the orbital plane In the subsequent expressions we neglect
all terms of rst or higher order in the small angle resp
0 0
2
a
0
2
a a a a
dr p y Rock y 0 y 2 y
B C
a p
A
Rock y 0
sign
0
B C
p sign
A
2 0
b
0
a a a a
dr p y Rock y 0 y 2 y
cos u u
0
B C
sin u u a p
A
0 Rock y 0
p
2 B C
q
A
0
2
2
sin u u
0 sin u u
0
0
In the rst case the p erturbation parallel to the direction sun satel lite is a
0
2
constant out of plane acceleration In the second case this p erturbation lies in the
0
orbital plane and oscillates Both the R and S comp onents oscillate with a p erio d equal to
the revolution p erio d of the satellite
the p erturbation due to the y bias is a constant along track In the rst case
0
2
acceleration the sign dep ending on the p osition of the sun In the second case
0
this p erturbation consists of an along track and an out of plane comp onent The out of
plane comp onent is constant in absolute value but changes sign for u u and u u
0 0
The along track comp onent is zero everywhere except near u u and u u where the
0 0
acceleration reaches p in absolute value Obviously we have in this case a strong correlation
2
with the pseudo sto chastic pulses
Approximate Perturbation Equations for Almost Circular Orbits
In Beutler we nd the p erturbation equations for the Keplerian elements semima jor
axis a numerical eccentricity e inclination i right ascension of the ascending no de
argument of p erigee and mean anomaly at time t
0
p
p
a e sin v R S
2
r
n e
p
2
e
e sin v R cos v cos E S
n a
r cos v
()
p
W i
2 2
n a e
r sin v
p
W
2 2
n a e sin i
p
2
r e
cos v R sin v S
e n a p
cos i
2
e r
R cos v e
e n a p
r n
sin v S t t a
0
p a
where r is the length of the radius vector at time t n is the mean motion of the satellite
2
E is the eccentric anomaly v the true anomaly and p a e the parameter of the
ellipse R S and W are the accelerations in radial tangential actually normal to R in
the orbital plane and out of plane directions
GPS orbits are low eccentricity orbits If we neglect all terms of the rst and higher orders
in the eccentricity e circular orbit these equations may b e considerably simpli ed We may
formally assume the p erigee to coincide with the ascending no de at time t which allows
0
us to use the approximation v E u where u n t t Moreover we do not need
0
to consider the equation for the argument of p erigee in this case We include instead an
equation for the argument of p erigee u at time t by adding the equations for the argument
0 0
of p erigee and for the mean anomaly at time t The resulting p erturbation equations
0
are
S a
n
sin u R cos u S e
n a
cos u
()
i W
n a
sin u
W
n a sin i
n
u R cos i t t a
0 0
n a a
Let us assume that the initial values for these elements at time t are a e i and u
0 0 0 0 0 00
The in uence of the p erturbation terms due to radiation pressure at time t are computed
by the equations a a t a e e t e i i t i t and
0 0 0 0
u u t u Usually the dominant p erturbation term is that in along track direction
0 0 00
i e the error in the argument of latitude u t at time t
u t u t n t t
0 0
where using Kepler s third law we have
n
a n
a
The solutions of the ab ove simpli ed p erturbation equations formally may b e written as
follows
Z
t
0
a t S dt
n
t
0
Z
t
0
e t sin u R cos u S dt
n a
t
0
Z
t
0
i t cos u W dt
n a
t
0
Z
t
0
t sin u W dt
n a sin i
t
0
Z
t
0
R dt cos i t u t
0
n a
t
0
Z
t
n n
0 0
t t a a t dt
0
a a
t
0
This last equation allows us to write the equation for the argument of latitude in the more
explicit form
Z Z
t t
n
0 0 0
R dt cos i t a t dt u t
n a a
t t
0 0
Solution of the Perturbation Equations for Radiation Pressure Terms
Let us deal separately with the terms p and p and with the cases when the sun lies in the
0 2
orbital plane resp when the sun is in the zenith of the orbital plane
0 0
2
Perturbations due to p Sun in Zenith of Orbital Plane
0 0
2
The p erturbing acceleration reads as
R
B C B C
S
a p
A A
0 y 0
W sign
0
We easily see that there are only short p erio d p erturbations in the orbital elements i and
The result of the integration is easily veri ed to b e
p sign
0 0
cos u t
2
n a sin i
p sign
0 0
i t sin u
2
n a
The p erturbations are purely p erio dical where the p erio d is equal to the revolution p erio d
of the satellite Obviously a small error in the parameters p will not contaminate the
0
estimation of earth rotation parameters This statement is no longer valid if we are lo oking
for a sub diurnal resolution of these terms In this case a correlation of the p erturbation
in with length of day estimates is very well p ossible For GPS satellites semima jor axis
0 6 2
a k m revolution p erio d U s p m s we have
0
t mas sign cos u
0
i t mas sign sin u
0
or in meters
a t m sign cos u
0
a i t m sign sin u
0
The p erio dic variations thus are of a considerable size
Perturbations due to p Sun in Orbital Plane
0 0
The p erturbing acceleration reads as
R cos u u
0
B C B C
S p sin u u a
A A
0 0 0 y
W
In this case there obviously is no out of plane comp onent and no correlation with earth
rotation parameters has to b e exp ected On the other hand there are strong R and S com
p onents One easily veri es that these accelerations generate periodic perturbations p erio d
equal to orbital p erio d amplitudes comparable to those in the previous section in a u
0
and u t The prominent p erturbation feature however is the p erturbation in the eccentri
city e where we have p erturbations prop ortional to sin u Because u has an annual p erio d
0 0
revolution of the sun around the earth or vice versa the p erio d of this p erturbation will
b e roughly one year to o Its amplitude is of the order of a few kilometers Beutler
Perturbations due to p Sun in Zenith of Orbital Plane
2 0
2
The p erturbing acceleration reads as
R
B C B C
a S p sign
A A
2 y 2 0
W
We merely have a constant along track component S of the p erturbing acceleration Periodic
variations p erio d revolution p erio d amplitudes comparable to those in and i given
ab ove in e will b e one consequence
The prominent p erturbation will b e that in a and in u t however We include these
results although a correlation with UT UTC is not expected in this case The quadrature
of the p erturbation equation in the semi ma jor axis a may b e p erformed without problems
a t p t t sign
2 0 0
n
This p erturbation in a pro duces a very pronounced p erturbation in along track direction
2
p t t sign a u t
2 0 0
where we have left out all p erio dic p erturbations
10 2 10
Typically the y bias p is of the order of a few m s Assuming a value of p
2 2
2
m s this gives rise to an along track e ect of ab out meters in along track direction
Theoretically there might b e a correlation of an orbit error in along track direction S with
the UT UTC drift This correlation would b e for low inclination satellites Because GPS
o
satellites have inclination of and b ecause with days our arc length is usually quite
long we do not exp ect a signi cant contribution in this case
Perturbations due to p Sun in Orbital Plane
2 0
The p erturbing acceleration reads as
R
q
B C B C
2
2
S p a
A A
sin u u
2 2 y
0 0
0
W
sign sin u u
0
Let us start our considerations by lo oking at the along track comp onent S This comp onent
is very small everywhere but in the immediate vicinity of these instants of time where
u t u and u t u For u u and u u the S comp onent reaches its
0 0 0 0
maximum value which is equal to j p j The total e ect thus actually corresp onds to a
2
velocity change at the time when u u resp u u Keeping in mind that p is of
0 0 2
10 2
the order of a few m s which acts only for a very short p erio d of time we conclude
that the along track e ect is very small when compared to the same e ect when the sun is
in the zenith of the orbital plane compare previous section We will not further consider
this e ect
The out of plane comp onent wil l in general pro duce a net rotation of the orbital plane
This b ecomes apparent if we introduce the ab ove R acceleration into the p erturbation
equation for
Z
t
p
2
0
t sin u sign sin u u dt
0
n a sin i
t
0
The worst case obviously results for u or u which will b e the case in spring and
0 0
fall Let us give the result for u
0
Z
t
p
2
0
j sin u j dt t
n a sin i
t
0
It is thus easy to verify that the net rotation after one revolution is
p
2
t U
0
2
n a sin i
10 2
For a typical y bias of p m s a rotation ab out mas p er day is the result
2
This seems to b e a very small value but it is not negligible From Figure we conclude
that we are lo oking for an e ect of ab out mas day The conclusion is thus clear a
correlation of the y biases with the drift parameter in UT UTC cannot b e excluded Let
us lo ok for empirical evidence in the next section
RESULTS FROM GLOBAL IGS DATA PROCESSING USING DIFFERENT
ORBIT STRATEGIES
The IGS Data and Orbit Strategies Used
To study the correlation b etween the y bias and the UT UTC drift days of data from
the global IGS network were pro cessed using di erent options see b elow The time p erio d
from day to day in was selected b ecause it contains b oth a p erio d where
no satellites were eclipsed at the b eginning of the series and a p erio d with satellites
PRN and going through the earth shadow once p er revolution
from the middle to the end of the series The data from ab out IGS sites were included
Overlapping days solutions as in the routine CODE analysis were pro duced for the
p erio d mentioned ab ove using the following orbit estimation strategies
Strategy A No pseudo sto chastic pulses were set up one y bias and one direct radiation
pressure parameter estimated on top of the Ro ck mo dels
Strategy B Pseudo sto chastic pulses were set up in R and S direction for the eclipsing
satellites only The two pulses p er revolution were set up ab out minutes after the
shadow exit of the satellite Y bias and direct pressure parameters as in the strategy
A
Strategy C Pseudo sto chastic pulses were set up in the R and S direction for al l satellites
once p er revolution at h and h
Strategy D Pulses were set up as in solution C but for all the three comp onents R S W
Strategy E Same solution as C but the y bias parameters were constrained using an a
10 2
priori sigma of m s to the values determined from years of y bias results
Strategy F Same solution as C but instead of estimating a y bias for each satellite and
arc a constant along track acceleration was determined The same constraints were
applied to the along track acceleration as in the strategy E to the y bias
In all these solutions trop osphere parameters p er day and site were set up and orbital
elements Keplerian elements and the parameters p and p see eqn or for solution of
0 2
type F a constant along track acceleration were determined The x and y p ole co ordinates
and UT UTC were estimated exactly in the same way as in the routine pro cessing CODE
Rothacher et al
UT UTC Results and Comparison to the Rotation of the Satelli te No des
The UT UTC drifts of the middle days of the days solutions were summed up to give
GPS derived UT UTC series The series resulting from strategies A through F are shown
in Figure The gure clearly indicates that the orbit estimation strategy has an imp ortant
impact on the UT UTC drift Strategies E and F exhibit the smallest drifts at least for
this time p erio d
Figure UT UTC series estimated from global days solutions using di erent orbit es
timation strategies
To see whether the p erturbations due to p may explain the UT UTC series in Figure
2
the p erturbation eqn was used to compute the rotation of the ascending no de due
to p for each individual satellite The mean value of these daily rotations over all satellites
2
should then re ect a mean rotation of the reference frame that may b e absorb ed by a drift
in UT UTC The actual computation was p erformed by numerically integrating eqn
using eqn for the out of plane p erturbing force W and the y bias values that were saved
for each individual solution The resulting mean rotation of the no des is shown in Figure
for all strategies except F Because the y bias was xed to the same a priori values as in
solution E the rotation of the no des would b e very similar to the ones of type E
Figure Mean rotation of the no des computed from the y bias parameters estimated using
di erent orbit strategies
Comparing Figures and we see the high correlation b etween the UT UTC series
and the mean rotations of the no des This shows that the estimated y bias plays indeed
a key role in the UT UTC series originating from GPS Although most of the features
visible in the UT UTC series may b e explained by the correlation with the y bias estimate
discussed ab ove there still remain residual drifts in UT UTC when compared to VLBI
The remaining drift will have to b e studied in future
Orbit Quality of the Strategies Used
From Figures and we see that constraining the y biases solution type E or estimating a
constant along track acceleration solution type F instead of a y bias improves the stability
of the UT UTC estimates It is therefore imp ortant to check whether the orbit quality is
the same in these two cases compared to e g solution type C Figure summarizes the rms
errors obtained by tting a days arc through the daily solutions of GPS week using
the radiation pressure mo del describ ed in Beutler et al
Figure Orbit quality assessment by tting a days arc through the daily solutions
The small degradation of the orbit quality for solutions E and F with resp ect to solutions
types C and D indicates that there remain some unmo deled biases in the orbits if the y biases
are heavily constrained or not estimated at all The estimation of a constant along track
acceleration is not su cient to get down to the consistency level obtained with strategies
C and D It is interesting to note that the estimation of pseudo sto chastic pulses in the W
direction do es not signi cantly change the UT UTC b ehaviour
SUMMARY AND CONCLUSIONS
In the rst part of this article we developed the p erturbations due to the radiation pressure
parameters p and p using sp ecial p erturbation theory We showed that the nature for these
0 2
p erturbations is quite di erent for the cases when the sun is in the zenith of the orbital plane
resp lies in the orbital plane We have shown in particular that a net rotation of the orbital
plane around the axis p ointing to the celestial p ole will result due to a constant y bias p
2
in the case where the sun lies in the orbital plane Such rotations are potentially dangerous
b ecause they are very closely correlated with the estimates for the drifts in UT UTC
We found evidence that our theoretical considerations are relevant in practice Figures
and underline this statement Since June solution of Type C is the o cial solution of
the CODE Analysis Center We are in the pro cess of studying solutions of type E in order
to reduce the bias in our UT UTC drift estimates Our concern at present is a degradation
of the orbit quality as seen in Figure
An alternative which is b elieved to b e sup erior consists of removing the Ro ck mo dels
completely and to replace them by the new radiation pressure mo del developed in Beutler
et al An analysis of the kind presented in this pap er will have to b e p erformed for
this new mo del to o
We should state in conclusion however that length of day estimates made by GPS will
always su er from p otential correlations with orbit parameters which have to b e estimated
This situation could b e changed radically if direction observations of highest quality of the
order of a few mas were available for GPS satellites
REFERENCES
Beutler G Himmelsmechanik II Der erdnahe Raum Mitteilungen der Satelliten
Beobachtungsstation Zimmerwald No Druckerei der Universit atBern
Beutler G E Bro ckmann W Gurtner U Hugentobler L Mervart M Rothacher
Extended Orbit Modeling Techniques at the CODE Processing Center of the Interna
tional GPS Service for Geodynamics IGS Theory and Initial Results Manuscripta
Geo daetica Vol pp
Fliegel H F T E Gallini E R Swift Global Positioning System Radiation Force
Model for Geodetic Applications Journal of Geophysical Research Vol No B
pp
Rothacher M R Weber U Wild A Wiget C Boucher S Botton H Seeger An
nual Report of the CODE Processing Center of the IGS International GPS Ser
vice for Geo dynamics Annual Rep ort ed J Z Zumberge R Liu and R Neilan
JPL California in preparation