J Geod (2012) 86:309–317
DOI 10.1007/s00190-011-0517-4
ORIGINAL ARTICLE
Impact of Earth radiation pressure on GPS position estimates
C. J. Rodriguez-Solano · U. Hugentobler ·
P. Steigenberger · S. Lutz
source: https://doi.org/10.7892/boris.17848 | downloaded: 31.5.2020
Received: 31 March 2011 / Accepted: 20 September 2011 / Published online: 13 October 2011
© Springer-Verlag 2011
Abstract GPS satellite orbits available from the International GNSS Service (IGS) show a consistent radial bias of
up to several cm and a particular pattern in the Satellite Laser
Ranging (SLR) residuals, which are suggested to be related
to radiation pressure mismodeling. In addition, orbit-related
frequencies were identified in geodetic time series such as
apparent geocenter motion and station displacements derived
from GPS tracking data. A potential solution to these discrepancies is the inclusion of Earth radiation pressure (visible and
infrared) modeling in the orbit determination process. This
is currently not yet considered by all analysis centers contributing to the IGS final orbits. The acceleration, accounting for Earth radiation and satellite models, is introduced
in this paper in the computation of a global GPS network
(around 200 IGS sites) adopting the analysis strategies from
the Center for Orbit Determination in Europe (CODE). Two
solutions covering 9 years (2000–2008) with and without
Earth radiation pressure were computed and form the basis
for this study. In previous studies, it has been shown that
Earth radiation pressure has a non-negligible effect on the
GPS orbits, mainly in the radial component. In this paper,
the effect on the along-track and cross-track components is
studied in more detail. Also in this paper, it is shown that
Earth radiation pressure leads to a change in the estimates
of GPS ground station positions, which is systematic over
large regions of the Earth. This observed “deformation” of
the Earth is towards North–South and with large scale patterns that repeat six times per GPS draconitic year (350 days),
C. J. Rodriguez-Solano (B) · U. Hugentobler · P. Steigenberger
Institut für Astronomische und Physikalische Geodäsie,
Technische Universität München, 80333 Munich, Germany
e-mail: rodriguez@bv.tum.de
S. Lutz
Astronomisches Institut, Universität Bern, 3012 Bern, Switzerland
reaching a magnitude of up to 1 mm. The impact of Earth
radiation pressure on the geocenter and length of day estimates was also investigated, but the effect is found to be less
significant as compared to the orbits and position estimates.
Keywords GPS · Albedo · Precise orbit modeling · Spectra
of GPS position estimates
1 Introduction
The International GNSS Service (IGS, Dow et al. 2009) final
orbits have reached an internal precision of around 2.5 cm for
GPS and 5 cm for GLONASS satellites1 from a level about
an order of magnitude larger in the mid 1990s. The progress
can be attributed to understanding the errors and improving
the models affecting the GNSS technique, including those
related to precise orbit modeling. Despite the performance
attained, however, some problems remain in the orbits but as
well as in the position estimates of GPS tracking stations.
The orbits of the two GPS satellites equipped with laser
retro reflector arrays (LRA) show a consistent radial bias of
up to several cm, when compared with the Satellite Laser
Ranging (SLR) measurements, known as the GPS – SLR
orbit anomaly. This bias was observed for the Center for
Orbit Determination in Europe (CODE)2 final orbits with a
magnitude of 3–4 cm by Urschl et al. (2007). More recent
1 http://igscb.jpl.nasa.gov/components/prods.html, accessed on 25
August 2011.
2
CODE is a consortium formed by: the Astronomical Institute of the
University of Bern (AIUB, Bern, Switzerland), the Swiss Federal Office
of Topography (swisstopo, Wabern, Switzerland), the Federal Agency
for Cartography and Geodesy (BKG, Frankfurt am Main, Germany),
and the Institut für Astronomische und Physikalische Geodäsie of the
Technische Universität München (IAPG, Munich, Germany).
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mination process. Two solutions covering nine years (2000–
2008) with and without Earth radiation pressure were computed (Sect. 3). This allows to study the impact on the GPS
orbits (mainly focussing on the along- and cross-track components, Sect. 4) as well as on GPS derived geodetic parameters like estimated station positions (Sect. 5), geocenter
coordinates and length of day (Sect. 6).
2 Models
Fig. 1 Relative geometry of Sun, Earth and GPS satellite
comparisons between GPS and SLR measurements (for various IGS analysis centers, Bar-Sever et al. 2009) show a
smaller bias of 12–22 mm with associated scatters of 16–
25 mm. These figures can be used as a measure of the current
radial accuracy of IGS final orbits. In addition, orbit related
frequencies were identified in geodetic time series such as
apparent geocenter motion by Hugentobler et al. (2006) and
station displacements derived from GPS tracking data by
Ray et al. (2008). Particularly, an anomalous frequency of
1.04 cpy (cycles per year) was found, corresponding to a
period of about 350 days which is very similar to the GPS
draconitic year, the repeat period of the Sun with respect to
the satellite constellation.
Urschl et al. (2007) plotted the GPS–SLR residuals as a
function of the position of the Sun with respect to the satellite,
more specifically in a (β0 , u) reference frame, where β0 is
the elevation angle of the Sun above the orbital plane and u
is the argument of latitude of the satellite with respect to the
argument of latitude of the Sun (see Fig. 1). The plot of Urschl
et al. (2007) shows a peculiar pattern indicating GPS orbit
modeling deficiencies, and the radiation pressure caused by
the Earth albedo (not considered at that time in the GPS orbit
determination) was mentioned as one of the possible causes
to be investigated. Moreover, since the orbit perturbations
caused by Earth radiation pressure depend on the position of
Sun, Earth and satellite, neglecting Earth radiation pressure
is also a good candidate for causing the observed anomalous
frequencies in the geodetic time series.
The mathematical formulation used in our study for the
computation of Earth radiation pressure (Sect. 2.2) is based
on the model of Knocke et al. (1988). The construction of
the satellite models for the interaction with Earth radiation
(Sect. 2.3) is based on the information contained in Fliegel
et al. (1992) and Fliegel and Gallini (1996), who developed solar radiation pressure models for GPS satellites. More
recently, Ziebart et al. (2007) and Rodriguez-Solano et al.
(2010) studied the impact of Earth radiation pressure on GPS
orbits (mainly in the radial component) and on GPS–SLR
residuals (see also Sect. 4). In the current paper, we include
the acceleration due to Earth radiation pressure (visible and
infrared, Sect. 2.4) acting on GPS satellites in the orbit deter-
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2.1 Selection of models
In a previous work (Rodriguez-Solano et al. 2010), Earth
radiation and GPS satellite models of increasing complexity
were developed and tested. From that study, the key factors
of the models have been identified. For the Earth radiation, it
was found that the use of an analytical model (based on constant albedo) or one based on numerical integration of Earth’s
actual reflectivity and emissivity data, give similar results for
the irradiance acting on the GPS satellites (difference of up to
10%), mainly due to the high altitude of these space vehicles.
As data from the Clouds and Earth’s Radiant Energy System
(CERES, Wielicki et al. 1996) satellite mission are available since the beginning of 20003 , the numerical approach
was chosen (see Sect. 2.2). Concerning the satellite model,
use of a box-wing model is a key factor since the variation
of the solar panels with respect to the Earth is very important, whereas a simple cannon-ball model is highly inaccurate. The accuracy of the modeled optical properties and the
detailed structure of the satellite have a lower impact on the
acceleration than the use of a box-wing model, but they are
still important enough to be considered (see Sect. 2.3). Further details of the models may be found in Rodriguez-Solano
(2009).
2.2 Earth radiation
The mathematical formulation of the Earth radiation model
used for this study is the same as the one proposed by
Knocke et al. (1988) for computing the irradiance received
by the satellite due to the Earth’s reflected (visible) and emitted (infrared) radiation. It is assumed that the Earth reflects
and emits radiation in a purely diffuse way as a Lambertian
sphere. The main steps included in the model are: (1) Determine the solar irradiance received by each surface element
of the Earth (grid of 2.5◦ × 2.5◦ ). (2) Compute the irradiance
received by the satellite based on the reflectivity and emissivity coefficients (from NASA’s CERES project, also used
by Ziebart et al. 2004 but not yet by Knocke et al. 1988) for
3
http://eosweb.larc.nasa.gov/PRODOCS/ceres/level3_es4_table.
html, accessed on 29 March 2011.
Impact of Earth radiation pressure on GPS position estimates
311
each Earth’s surface element. (3) Compute the Earth radiation pressure caused by the interaction between the irradiance from each surface element of the Earth with the satellite
model.
2.3 GPS satellites
The physical description of the interaction between radiation
and the surfaces of the satellite is provided by Fliegel et al.
(1992). It is based on the optical properties of the surface,
e.g., specularity and reflectivity or equivalently the fraction
of reflected, absorbed and diffused photons which should
sum up to one (Milani et al. 1987). The dimensions and optical properties are given for Blocks I and II/IIA GPS satellites
in the mentioned paper and for Block IIR satellites in Fliegel
and Gallini (1996). These two papers are the basis for constructing our box-wing satellite model. They also provide a
priori models (ROCK) for solar radiation pressure for precise geodetic applications. However, nowadays no a priori
model or purely empirical models are used due to the lower
performance of the ROCK models compared to the empirical
models. For this study, the CODE empirical model for solar
radiation pressure (see also Sect. 3, Beutler et al. 1994) was
used with no a priori model.
To complete the box-wing model, the nominal attitude law
of the satellite must be considered, i.e., ensuring that the navigation antennas always point to the geocenter and that the
solar panels always point to the Sun. The nominal attitude
law is correct in most cases. Only when the satellite is in
eclipse season and in the Earth’s shadow as at orbit noon , is
this no longer true (Bar-Sever 1996; Kouba 2009). However,
non-nominal orbit noon and midnight maneuvers are not yet
considered in our model. With the assumption of nominal
attitude, the Earth radiation and satellite models have a main
dependency on the angle ψ formed by satellite, Earth and
Sun, as shown in Fig. 1, which can be simply written as
cos(ψ) = cos(β0 ) cos(u).
Finally, the thrust of the navigation antennas, as reported
by Ziebart et al. (2004), was also included in the satellite
model. An approximate value of 80 Watts of antenna transmission power was used for all GPS satellites (Block II/IIA
and Block IIR). In previous studies (Ziebart et al. 2004;
Rodriguez-Solano et al. 2010), the thrust of the navigation
antennas was found to cause a non-negligible effect for GPS
satellite orbits. It introduces a constant radial acceleration of
around 2.7 × 10−10 m/s2 for Block IIA satellites, a magnitude which is comparable to the minimum radial acceleration
due to Earth radiation pressure (around 4 × 10−10 m/s2 ).
2.4 Acceleration
By combining the irradiance from each surface element of the
Earth with the satellite model, a force acting on the satellite is
Fig. 2 Acceleration acting on Block IIA satellites due to Earth radiation pressure plus antenna thrust, in a Sun-fixed reference frame. When
u equals 180◦ , the satellite is above the shadowed side of the Earth
obtained. By integrating these forces over the part of the Earth
visible to the satellite and dividing by the mass of the satellite,
the acceleration due to Earth radiation pressure for a specific
satellite position is obtained. In Fig. 2, the resulting acceleration is shown for a box-wing GPS Block IIA satellite model
in a Sun-fixed reference frame (β0 , u) for the radial, alongtrack and cross-track components. To produce this plot, an
analytical Earth radiation model was used assuming a globally constant albedo of 0.3 as well as only a radial impact
direction of the irradiance reaching the satellite. This was
done for computational efficiency. The acceleration used to
calculate the GPS satellite orbits was, however, based on the
numerical Earth radiation model (with CERES satellite data,
Sect. 2.2).
For the radial component of the acceleration (Fig. 2), the
maximum at u = 0◦ and β0 = 0◦ corresponds to the point
where Sun, satellite and Earth are exactly aligned, with the
spacecraft above the Sun-facing side of the Earth. Note that
for 90◦ < u < 270◦ the satellite is mainly above the night
hemisphere of the Earth. At u = 180◦ and β0 = 0◦ it
is in total shadow, where one finds a secondary maximum
of the radial acceleration since the solar panels are maximally exposed to the infrared radiation of the Earth. This last
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feature would not be present for a cannon-ball model with
constant cross-section, since the solar panels change their
orientation a lot with respect to the Earth over one revolution. This change of orientation is also responsible for the
minima at u = 90◦ and u = 270◦ , where the geocentric
directions to the satellite and to the Sun are orthogonal and
the exposure of the solar panels to Earth radiation is almost
zero.
The acceleration observed in the along-track and crosstrack components in Fig. 2 is an effect of the solar panels;
i.e., with a cannon-ball model we would not see these particular patterns. Moreover, the along-track acceleration has
a period of twice per revolution while the cross-track period
is only once per revolution. Also interesting is the change of
sign in the cross-track acceleration with the sign of the β0
angle.
3 Processing strategy
The Earth radiation pressure model as described in Sect. 2.2
and 2.3 was implemented in the Bernese GPS Software (Dach
et al. 2007). Adapting the strategy of CODE as described
by Steigenberger et al. (2006, 2011), 9 years (2000–2008)
of GPS ground tracking data were processed, using around
200 IGS stations. Two solutions were computed, one including Earth radiation pressure and one without. No net rotation
(NNR) and no net translation (NNT) conditions were applied
with respect to the IGS05 (Ferland 2006) reference frame for
a subset of up to 125 tracking stations. The last condition is
required to estimate the offset between the Earth’s center of
mass, as sensed by the satellites, and the reference frame
origin. The resulting orbits and station coordinates are then
obtained in the terrestrial reference frame and the respective
comparisons (Sects. 4 and 5) are not affected by the apparent changes of the geocenter. Moreover, the impact of Earth
radiation pressure on the apparent geocenter can be studied
independently (Sect. 6).
The solar radiation pressure effect (in both solutions) was
modeled by estimating five empirical parameters (per satellite and per day) and with no a priori model. The five empirical parameters were proposed by Beutler et al. (1994) and
are basically the following: three constants in the D, Y and
B directions (see Fig. 1) and two periodic (once per revolution) in the B direction. Additionally, according to the
CODE processing strategy, three pseudo-stochastic pulses
once per revolution in the radial, along-track and cross-track
directions were estimated (individually constrained for each
direction). Despite having 9-year solutions, in most of the
plots presented in the following sections just the year 2007
is shown to keep the figures simple. The full 9-year solutions
are mainly used for computing the spectra of daily position
estimates and geocenter.
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Fig. 3 Top three RMS of GPS orbit differences per orbital plane (in
color) and per day of year for 2007. Bottom Sun elevation angle above
the GPS orbital planes (in color)
4 Impact on GPS orbits
The orbit differences presented in this section were obtained
by comparing the orbits of all GPS satellites computed with
and without Earth radiation pressure. The results in Figs. 3
and 4 are plotted in a local Earth-fixed reference frame
(North, East, height) rather than in an orbit-fixed reference
frame (radial, along-track, cross-track) in order to highlight
the relation of the orbit perturbations due to Earth radiation
pressure and the position estimates of GPS ground tracking
stations (see Sect. 5).
The most prominent effect of Earth radiation pressure on
the GPS orbits is a radial offset of 1–2 cm, observable in the
height component of Figs. 3 and 4. As already mentioned by
Ziebart et al. (2007), this effect reduces the GPS–SLR range
discrepancy by 1–2 cm, from which around 0.5 cm can be
attributed to the antenna thrust (Sect. 2.4, Rodriguez-Solano
et al. 2010). The reason for this radial reduction of orbits is
that GPS measurements, being essentially angular measurements due to required clock synchronization, mainly determine the mean motion of the satellite. As a matter of fact, a
Impact of Earth radiation pressure on GPS position estimates
Fig. 4 Mean orbit differences of all GPS satellites plotted as a function
of latitude and day of year (2007)
constant positive radial acceleration (equivalent to a reduction of GM, the product of the gravitational constant and
the mass of the Earth) decreases the orbital radius according to Kepler’s third law (Rodriguez-Solano et al. 2010).
From Fig. 3, one can also note the dependency of the height
RMS orbit differences with respect to the Sun elevation angle
above (or below) the orbital plane (β0 ). Consequently, this
kind of perturbation should have a main repeat period close
to half of the GPS draconitic year, that is about 350/2 days.
The subdaily dependency of the orbit differences on the u
angle was averaged when Figs. 3 and 4 were produced, since
the differences are presented as RMS and mean values per
day for the corresponding figures.
However, the North and East RMS orbit differences in
Fig. 3 do not seem to have a correlation with the β0 angle
of the respective planes, but rather to the combined effect
of the β0 angles of the six orbital planes. More specifically,
one finds a minimum in all the orbital planes, repeating six
times during 1 year, for the North component RMS orbit differences (with an exception for day 150 and for the orbital
plane E). These minima occur when the β0 angle of two orbital planes get close to zero degrees (shown with dotted lines
in Fig. 3), i.e., when the Sun gets close to the intersection line
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of the two orbital planes. From the acceleration produced by
Earth radiation pressure (Fig. 2), we know that the cross-track
acceleration is zero for β0 = 0◦ , therefore one could expect a
minimum in the North or East RMS orbit differences for the
two orbital planes with a Sun elevation angle close to zero.
However, the other four orbital planes with β0 angles different from zero also seem to be minimally affected by Earth
radiation pressure (mainly the North component) and have
minima at the same periods. The mechanism for this behavior is not yet understood, but it could be further investigated
by e.g. excluding the satellites from one or two orbital planes
in the solution computation.
By taking the mean of the orbit differences for all GPS
satellites for a given latitude (one degree intervals) and day
of year, one gets a very interesting result (Fig. 4). For example, the orbit height differences follow the position of the
Sun with respect to the Earth over the year, something that
is expected since the Earth radiation at the satellite reaches
its highest values if there is an alignment of Sun, satellite
and Earth. For the East component we find almost no signal
(compared to height and North) in the orbit differences. In
the North orbit differences we obtain a very similar pattern
as the one of Fig. 3, but in addition we observe that the differences are positive for the northern hemisphere and negative
for the southern one. This can be interpreted as a change in
the inclination of the orbits due to Earth radiation pressure.
In fact, the inclination changes by around 1 × 10−8 degrees,
which corresponds to 4.6 mm in the cross-track orbit positions (Fig. 4).
5 Impact on GPS station position estimates
Earth radiation pressure mismodeling causes a very particular perturbation at the millimeter level in the daily position
estimates of GPS ground tracking stations. Figure 5 shows
horizontal displacement vectors between solutions with and
without modeled Earth radiation pressure. This is shown
for 2 days of 2007, one where the North orbit differences
(see Fig. 3) are large (day 61) and one where they are minimal (day 91). For day 61, a “deformation” of the Earth in
north- and southwards direction is visible. Moreover, the two
characteristic patterns in Fig. 5 alternate six times per GPS
draconitic year (350 days). There are also differences in the
station height components (not shown), but their magnitude
is smaller than for the North component, which are the most
prominent. The comparison of the station positions based on
solutions with and without Earth radiation pressure, shown
in Fig. 5, was performed using a 7-parameter Helmert transformation. The scale difference is around 0.14 ppb (parts
per billion), or a uniform height shift of about 0.9 mm. In
Fig. 6, the translation parameters are shown. The shift in the
Z -axis is related to the distribution of GPS stations: as more
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Fig. 5 Change in the estimated horizontal GPS ground station positions (about 200 IGS sites) due to Earth radiation pressure for day 61 (left) and
day 91 (right) of year 2007
Fig. 6 Helmert translation parameters between GPS position solutions
with and without Earth radiation pressure for year 2007
stations are located in the northern hemisphere these dominate the no net translation condition imposed in the solution
(see also Sect. 3). Without the Helmert transformation we
would observe in Fig. 5, the northern stations were almost
unchanged and the southern ones were largely displaced. The
translation in the X -axis is accompanied by a rotation around
the same axis of about −2 µas, the rotations around Y and
Z are both within ±1 µas.
To demonstrate the correlation between the effects
observed in Sect. 4 and the ground station displacements,
we plot the mean of the North component differences (of
both orbits and stations) as a function of day of year and latitude (one degree intervals). Figure 7 shows basically the same
information as Fig. 4 but plotted in a different way: the differences are shown along the ordinate axis while colors indicate
the latitude. This representation is more appropriate to display station displacements as a plot similar to Fig. 4 would
leave large gaps in the southern hemisphere. The results show
the strong correlation between the orbit perturbations and the
station displacements. However, the magnitude of the orbit
differences is around five times larger than the station differences. If we imagine a GPS satellite placed almost directly
over a ground station, both in a middle northern latitude, the
satellite would be “pushed” by Earth radiation pressure northwards as well as the station. This relation is almost direct;
e.g., if we compute the ratio between the satellite’s semimajor
axis and the Earth radius (26,560 km/6,371 km) we obtain
a ratio of 4.17, very close to what we observe in Fig. 7. The
situation is of course much more complicated. Nevertheless,
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Fig. 7 North component of the mean stations differences (top) and
orbit differences (bottom) for year 2007, as a function of day of year
and color-coded latitude
we are then in a situation where a small change in the alongtrack or cross-track position of the GPS satellites (which can
not be absorbed by the clocks as in the case of the radial
component) leads to an almost direct effect on the position
estimates of GPS ground tracking stations.
Furthermore, the observed “deformation” of the Earth
changes the power spectrum of the North component of
GPS daily position estimates (shown in Fig. 8). This figure is based on nine years of tracking data from around 200
globally distributed IGS stations. By introducing Earth radiation pressure we obtain a reduction mainly in the sixth peak
of the power spectrum of the North component, located at
approximately 6 × 1.04 cpy. This frequency is also noticeable in Figs. 3, 4 and 7 (and in Fig. 5 if it would be animated), where we can see a systematic pattern of six cycles
over one GPS draconitic year (around 350 days), the repeat
period of the Sun with respect to the satellite constellation. As observed in Fig. 8 and also mentioned by Ray
et al. (2008), the sixth peak in the North component is
one of the sharpest and highest. This peak is reduced from
Impact of Earth radiation pressure on GPS position estimates
Fig. 8 Power spectrum of the North component of GPS daily position
estimates (around 200 IGS sites) from 2000 to 2008, with (blue) and
without (red) Earth radiation pressure
0.01443 mm2 to 0.00956 mm2 and after subtracting a noise
floor of 0.0016 mm2 , we obtain a variance reduction of 38%.
This result is very important since it indicates that the solution that includes Earth radiation pressure reduces systematically the anomalous frequency. It also demonstrates that the
observed anomalous frequencies in the GPS position time
series are related (at least partially) to orbit mismodeling
issues. However, the peaks in the East and height power
spectrum (not shown) do not exhibit a significant reduction.
The power spectra in this paper were not computed using
the Lomb–Scargle Periodogram (Scargle 1982; Press et al.
1992), as done by other authors (Ray et al. 2008; Tregoning and Watson 2009)4 . Instead, the power spectrum of the
Fast Fourier Transform (FFT, Press et al. 1992) was used
and where data is missing, zero padding was employed,
since GPS-derived daily positions (station coordinates and
geocenter) are evenly spaced. The units of the power spectrum are clear (mm2 ), while the Lomb–Scargle Periodogram (due to normalization) has no units, and finally, the
computation time is largely reduced when employing the
FFT. Additionally, to compute the mean power spectrum
of the 200 IGS stations (Fig. 8) a weighting according
to the inverse of the variance was introduced to ensure
that the noisier position time series have a lower contribution.
6 Impact on geocenter and LOD
The impact of Earth radiation pressure on other geodetic
parameters, specifically the geocenter position and the length
of day (LOD), was also investigated. For the geocenter we
find an impact at the one millimeter level (see Fig. 9), but
no clear pattern is visible as for the GPS position esti4
There are also other methods for computing the power spectrum, e.g.,
a robust estimation method was used by Collilieux et al. (2007) for identifying the draconitic harmonics in the individual time series of GPS
stations.
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Fig. 9 Geocenter differences between solutions with and without
Earth radiation pressure for year 2007
Fig. 10 Power spectrum of the Z -component of the GPS-derived geocenter from 2000 to 2008. The power spectrum difference between
the reference solution minus the one including Earth radiation pressure
multiplied by ten is shown in blue
mates (Sect. 5). The geocenter is appropriate for investigation since in the Z -component a period of 350 days,
the GPS draconitic year, corresponding to 1.04 cpy, was
identified by Hugentobler et al. (2006). The power spectrum of the geocenter time series (without modeling Earth
radiation pressure) is shown in Fig. 10. Note that the
main anomalous peaks are found at odd multiples (1, 3,
5, 7) of 1.04 cpy. By including the Earth radiation pressure a small reduction in the 5th and 7th peaks has been
achieved, 3.8 and 1.7%, respectively. Figure 10 shows the
difference of the power spectra of the solutions with and
without Earth radiation pressure, however multiplied by a
factor of 10. A negative sign of the difference, as found
for the 5th and 7th peaks, means a reduction of the original power spectrum, while a positive sign means an increase
as observed for the 1st and 3rd peaks (by 6.4 and 1.7%,
respectively).
Figure 11, finally, shows the difference in the LOD estimates obtained with and without Earth radiation pressure.
The impact is of the order of 10 µs with a pattern that has
a period close to 350 days, again, the GPS draconitic year.
However the LOD power spectrum derived from GPS measurements does not show significant peaks at harmonics of
1.04 cpy. Also the difference of the power spectra of the
LOD estimates only shows small variations at harmonics of
1.04 cpy.
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Fig. 11 Difference in length of day (LOD) between the solutions with
and without Earth radiation pressure from 2000 to 2008
7 Conclusions
The acceleration acting on GPS (i.e. box-wing-like) satellites caused by Earth radiation (visible and infrared) depends
mainly on the relative position of each satellite, Earth and
Sun. This implies a perturbation in the GPS satellite orbits
at harmonics of the GPS draconitic year. The main non-negligible effect on the orbits is a reduction of the height by
about 1–2 cm, consistent with the findings of Ziebart et al.
(2007). In the other components of the orbits (along-track
and cross-track), the effect is about one order of magnitude
smaller. When the orbit differences (with and without Earth
radiation pressure) are separated by orbital plane, it becomes
evident that when the Earth radiation pressure is zero in the
cross-track component for two orbital planes, the other four
orbital planes are also unaffected, the mechanism for which
requires further investigation.
Moreover, in the North component of the orbit differences
a particular pattern is found, displacing the orbits (few millimeters) towards North at northern latitudes and South at
southern latitudes, with a period of one sixth of the GPS
draconitic year. The same pattern is found in the daily position estimates of GPS ground stations at the sub-millimeter level, indicating that a small change in the along-track
or cross-track position of the GPS satellites (which can not
be absorbed by the clocks like the radial shift) leads to an
almost direct effect on the position estimates of GPS ground
stations. This “deformation” of the Earth also leads to a
reduction in the anomalous spectra of GPS daily position
estimates, mainly in the sixth peak of the North component
at 6 × 1.04 cpy.
The impact of Earth radiation pressure on the geocenter
(at the one millimeter level) is less significant. It reduces just
slightly some of the anomalous peaks of the geocenter power
spectrum and even increases others. For LOD, the impact is
of order 10 µs. The fact that the impact on geocenter and
LOD is low, indicates that other more important problems
remain in the orbit modeling, in particular for the geocenter. One of the most likely candidates is the solar radiation
pressure, which also has a strong dependency on the GPS draconitic year. It is larger in magnitude than the Earth radiation
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pressure and currently it is considered mainly by applying
empirical parameterizations.
Finally, the non-negligible effects of Earth radiation
pressure that have been found in the GPS orbits and station position estimates underlines the importance of further
improvements in the orbit-modeling techniques. Of particular concern is a better understanding and modeling of nonconservatives forces affecting the satellites, which involve
the modeling of the radiation source, the satellite structure
(including its attitude) and surface properties. This also justifies the efforts that are invested in this task by different
groups of scientists.
The subroutines to compute the acceleration due to Earth
radiation pressure are available at: http://www.iapg.bv.tum.
de/albedo/. The subroutines contain the Earth radiation models (analytical and numerical with CERES data) as well as the
box-wing models for GNSS satellites (GPS and GLONASS).
Acknowledgments The authors gratefully acknowledge the International GNSS Service (IGS, Dow et al. 2009) for providing the high
quality data needed for this study. The comments and suggestions by
Jim Ray (NOAA) and two other reviewers are greatly appreciated. We
acknowledge the support of the TUM Graduate School.
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