1. Modeling Earth radiation pressure Carlos Rodriguez-Solano
and its impact on
GPS orbits and ground tracking stations
Carlos Rodriguez-Solano
Urs Hugentobler
Peter Steigenberger
Tim Springer
Bernese GPS Software
NAPEOS Software

2. 1 Motivation GPS – SLR orbit anomaly: 4 – 5 cm
SLR residuals for GPS satellites (mean subtracted) in a Sun-fixed reference frame show a peculiar pattern: l
Angle satellite – Earth – Sun:
Urschl et al. (2008)

3. 1 Motivation More recently …
SLR range residuals based on reprocessed ESOC orbit series –
SLR and GPS agree very well!
Only a small bias (~1.8 cm) and eclipse season (attitude) effects remain

4. 1 Motivation
Orbit-related frequencies on geodetic time series  GPS draconitic year
Station coordinates
(> 200 IGS sites).
Also computed by:
Ray et al. (2009)
13.65 ± 0.02 days
Penna et al. (2007):
13.66 days
9 years of tracking data: –
Geocenter position.
Also pointed out by:
Hugentobler et al. (2006)

5. 2 Earth Radiation Model
Computation of Irradiance [W/m2] at satellite position, assuming:
Earth scattering properties approximated as a Lambertian sphere
emitted and reflected radiation  infrared and visible radiation
Types of models:
Analytical: Constant albedo, Earth as point source  only radial acceleration:
Numerical: Constant albedo, finite Earth radius
Latitude-dependent reflectivity and emissivity
Latitude-, longitude- and time-dependent reflectivity and emissivity
from NASA CERES project
AE = πRE2, RE = 6378 km, ESUN = 1367 W/m2, h = satellite altitude, α = albedo (≈ 0.3)

6. 2 Earth Radiation Model
CERES (Clouds and Earth's Radiant Energy System)
NASA EOS project
Reflectivity 
Emissivity 
CERES data, monthly averages, July 2007

7. 2 Earth Radiation Model Min. Diff.: Max. -3.2% +3.7% -6.7% +10.8%
E4: CERES data
(August 2007)
E3: Latitude dependency
E2: Numerical, constant albedo
E1: Analytical, constant albedo
Min.
Diff.:
Max.
-3.2%
+3.7%
-6.7%
+10.8%
-7.4%
+14.0%

8. 3 GPS Satellite Model Box-wing model Three main satellite surfaces:
1) +Z side, pointing always to the Earth
2) Front-side of solar panels, pointing always to the Sun
3) Back-side of solar panels
Main dependency on angle ψ satellite – Earth – Sun

9. 4 Acceleration on the Satellites
Earth radiation and satellite models of increasing complexity
for PRN06 and β0 = 20.2°
Along track acceleration [m/s2]
Radial acceleration [m/s2]
Cross track acceleration [m/s2]

10. 4 Acceleration on the Satellites
Key factors can be already identified:
- No large differences between Earth radiation models
- Analytical box-wing model with block specific optical properties and with antenna thrust
Most important factor  box-wing (solar panels change drastically w.r.t the Earth over one revolution)
Magnitude of acceleration compared to solar radiation pressure is just 1-2 %
But if the change of acceleration (minimum to maximum) is compared
 the effect is up to 20% of the solar radiation pressure
Solar radiation pressure  solar panels are fixed, bus changes orientation
Earth radiation pressure  bus is fixed, solar panels change orientation
Comparable to Y-bias effect (1x10-9 m/s2)

11. 5 Impact on the Orbits
Implementation of a priori acceleration in the Bernese GPS Software
Computation of GPS orbits as done by CODE for one year (2007) of tracking data
Orbit differences = perturbed orbit (with albedo) – reference orbit (without albedo)
PRN05
Simplest model
Earth radiation:
Analytical
GPS satellite:
Cannon-ball
PRN06

12. 5 Impact on the Orbits
Implementation of apriori acceleration in the Bernese GPS Software
Computation of GPS orbits as done by CODE for one year (2007) of tracking data
Orbit differences = perturbed orbit (with albedo) – reference orbit (without albedo)
PRN05
Most complex model
Earth radiation:
CERES data
GPS satellite:
Num. Box-Wing
Block specific
Antenna thrust
PRN06

13. 5 Impact on the Orbits
Orbit differences = perturbed orbit (with albedo) – reference orbit (without albedo)
Comparable with SLR – GPS residuals in a Sun-fixed reference frame (β0 and ∆u)
Urschl et al. (2008)

14. 5 Impact on the Orbits SLR validation: SLR measurements – GPS orbits
SLR-GPS orbit anomaly  mean reduction of 16 mm
- 1.1 cm  albedo (TUM, ESA)
cm  antenna thrust (TUM)
TUM:
ESA:

15. 6 Impact on the Ground Stations

16. 6 Impact on the Ground Stations
Change of spectra for the North coordinates,
> 200 IGS sites and 9 years of tracking data
Main reduction on the sixth peak
Where the other peaks come from?
 Solar radiation pressure?
Why this pattern on the North stations residuals?

17. 6 Impact on the Ground Stations …and Orbits
Orbit residuals  (NORTH) as a function of latitude and DOY
Mainly effect of cross-track component  orientation of solar panel
Almost direct effect of the orbits (cross-track) on the ground stations positions
Systematic “deformation” of the Earth

18. 7 Impact on the LOD
Change of Length of Day (LOD) due to Earth radiation pressure
 around 10 µs
Effect on other geodetic parameters
importance of orbit modeling

19. 8 Conclusions Earth radiation pressure has a non-negligible effect
 on GPS orbits (1x10-9 m/s2) comparable to Y-bias
 on ground stations (mainly North) at the submillimeter level
Albedo causes a mean reduction of the orbit radius of about 1 cm
The largest impact in periodic variations is caused by the solar panels
 Use of a box-wing satellite model is a must
Different Earth radiation models as well as satellite model details have a small impact on the orbits
Albedo can partially explain the peculiar pattern observed in SLR residuals
Recommendation for an adequate but simple modelling:
 Earth radiation model with CERES data (or alternatively the analytical model for constant albedo)
 Analytical box-wing model with block specific optical properties and with antenna thrust

20. 9 References
Fliegel H, Gallini T, Swift E (1992) Global Positioning System Radiation Force Model for Geodetic Applications. Journal of Geophysical Research 97(B1):
Fliegel H, Gallini T (1996) Solar Force Modelling of Block IIR Global Positioning System satellites. Journal of Spacecraft and Rockets 33(6):
Hugentobler U, van der Marel, Springer T (2006) Identification and mitigation of GNSS errors. Position Paper, IGS 2006 Workshop Proceedings
Knocke PC, Ries JC, Tapley BD (1988) Earth radiation pressure effects on satellites. Proceedings of AIAA/AAS Astrodynamics Conference:
Press W, Teukolsky S, Vetterling W, Flannery B (1992) Numerical Recipes in Fortran 77, 2nd edn. Cambridge University Press
Ray J, Altamimi Z, Collilieux X, van Dam T (2008) Anomalous harmonics in the spectra of GPS position estimates. GPS Solutions 12: 55-64
Rodriguez-Solano CJ, Hugentobler U, Steigenberger P (2010) Impact of Albedo Radiation on GPS Satellites. IAG Symposium – Geodesy for Planet Earth, accepted
Urschl C, Beutler G, Gurtner W, Hugentobler U, Schaer S (2008) Calibrating GNSS orbits with SLR tracking data. Proceedings of the 15th International Workshop on Laser Ranging: 23-26
Ziebart M, Sibthorpe A, Cross P (2007) Cracking the GPS – SLR Orbit Anomaly. Proceedings of ION- GNSS-2007:

21. 1 Motivation Consistent bias of 4 – 5 cm
 The GPS – SLR Orbit Anomaly.
Ziebart et al. (2007)

22. 1 Motivation Power Spectrum Estimation Using the FFT
Use of Discrete FFT instead of Lomb-Scargle periodogram
Why?
Data has the same time spacing (1 day) but problem with data missing
FFT still appropiate if data is missing and e.g. set to zero
Lomb-Scargle periodogram robust if time spacing is not the same, e.g. in astronomical measurements
As expected results are very similar using both methods
 but Power Spectrum using FFT is much faster and simpler
Press et al. (1992)

23. 1 Motivation

24. 1 Motivation
Period:
27.6 +/- 0.1 days

25. 2 Earth Radiation Model
Comparison of analytical and numerical models for constant albedo:
- Different albedos of the Earth
only emission
only reflection

26. 2 Earth Radiation Model
Comparison of analytical and numerical models for constant albedo:
- Different satellite altitudes

27. 2 Earth Radiation Model
E3 – E4
E2 – E4
E1 – E4

28. 3 GPS Satellite Model
General radiation pressure model from Fliegel et al. (1992,1996)
Analytical model assuming Earth radiation to be purely radial
 Acceleration acting on the satellites
Satellite Bus
Solar Panels
A: area of satellite surface ψ: angle satellite – Earth – Sun
M: mass of satellite μ: specularity, 0 diffuse to 1 specular
E: Earth‘s irradiance ν: reflectivity, 0 black to 1 white
c: velocity of light in vacuum

29. 4 Acceleration on the Satellites
Simpler model: cannon-ball model (no solar panels)  average over ψ
More sophisticated model: Numerical box-wing model
 considering the full disc of the Earth (not purely radial radiation)
In total three GPS satellite models:
- S1: cannon-ball
- S2: analytical box-wing
- S3: numerical box-wing
Additionally consideration of:
- B: block specific dimensions and optical properties
- A: thrust due to navigation antennas
Many possibilities: 4 Earth radiation models
3 GPS satellite models
2 extras (turn on/off)

30. 4 Acceleration on the Satellites
Earth radiation and satellite models of increasing complexity
for PRN06 and β0 = 20.2°
Along track acceleration [m/s2]
Radial acceleration [m/s2]
Cross track acceleration [m/s2]

31. 4 Acceleration on the Satellites
Earth Radiation Models:
E1: analytical, constant albedo
E2: numerical, constant albedo
E3: numerical, latitude dependent albedo
E4: numerical, CERES data
Other options:
B: block specific dimensions and optical properties
A: thrust due to navigation antennas
R: a priori solar radiation pressure (ROCK) model
GPS Satellite Models:
S1: cannon-ball
S2: analytical box-wing
S3: numerical box-wing

32. 4 Acceleration on the Satellites
Cannon-ball: radial acceleration
Box-wing: radial acceleration
4 Acceleration on the Satellites
Acceleration over one year in a sun-fixed coordinate system, E1-S1 and E1-S2
Minimum at dark side of the Earth
Maximum at dark side of the Earth
 Caused by infrared radiation acting on solar panels

33. 4 Acceleration on the Satellites
Acceleration over one year in a sun-fixed coordinate system, E1-S2
Box-wing: along track acceleration
Twice per revolution
Box-wing: cross track acceleration
Once per revolution

34. 4 Acceleration on the Satellites
Earth radiation pressure [m/s2]
From 0.5x10-9 to 2.5x10-9
Solar radiation pressure [m/s2]
From 9.5x10-8 to 10.5x10-8

35. 5 Impact on the Orbits
Orbit differences = perturbed orbit (with albedo) – reference orbit (without albedo)
/ / /
/ / /
/ / /
/ / /

36. 5 Impact on the Orbits
Orbit differences  effect of different models, PRN05
Num. (const. albedo) model
Box-wing analytical model
Latitude dependent albedo
CERES data

37. 5 Impact on the Orbits
Orbit differences  effect of different models, PRN05
Block specific properties
Box-wing numerical model
Antenna thrust

38. 5 Impact on the Orbits SLR validation: SLR measurements – GPS orbits
SLR-GPS orbit anomaly  mean reduction of 16 mm
- 11 mm  albedo
- 5 mm  antenna thrust
ITRF05
Scale parameter: / mm/Km
Comparison SLRF2005 and ITRF05RS
Red: with a priori ROCK model
Blue: no a priori ROCK model

39. 5 Impact on the Orbits

40. 5 Impact on the Orbits

41. 6 Impact on the Orbits

42. 6 Impact on the Orbits

43. 6 Impact