Algorithm Theoretical Basis Document

ESA AO/1-6367/10/NL/AF

“GOCE+ Theme 3: Air density and wind retrieval using GOCE data”

Version 1.1

Eelco Doornbos – TU Delft Pieter Visser – TU Delft Georg Koppenwallner – HTG Bent Fritsche – HTG

June 20, 2013 2 Contents

1 Introduction 7 1.1 About the project ...... 7 1.2 Purpose of this document ...... 7

2 Overall data ﬂow 9 2.1 Data processing ﬂow chart ...... 9 2.2 Input products ...... 9 2.2.1 EGG CCD (common-mode accelerations) ...... 9 2.2.2 EGG IAQ (attitude quaternions) ...... 11 2.2.3 SST PSO (precise science orbits) ...... 11 2.2.4 AUX NOM (ion engine thrust levels) ...... 11 2.2.5 GOCE mass ...... 11 2.3 Space environment models ...... 11 2.3.1 NRLMSISE-00 ...... 11 2.3.2 HWM07 ...... 12 2.3.3 Planetary ephemerides JPL DE405 ...... 13 2.4 Auxiliary input data ...... 13 2.4.1 Solar and geomagnetic activity data ...... 13 2.4.2 Earth orientation parameters ...... 14 2.5 Output products ...... 14

3 Data calibration 15 3.1 Accelerometer and ion engine calibration using GPS data ...... 15

4 Non-gravitational force modeling 25 4.1 GOCE satellite geometry model ...... 25 4.1.1 Modelling approach ...... 25 4.1.2 Modelling GOCE ...... 28 4.1.3 Reference frames ...... 29 4.2 ANGARA two-phase approach ...... 29 4.2.1 Aerodynamic analysis, Phase 1 ...... 30

3 4.2.2 Aerodynamic analysis, Phase 2 ...... 31 4.2.3 Radiation pressure analysis, Phase 1 ...... 31 4.2.4 Direct radiation pressure, Phase 2 ...... 32 4.2.5 Indirect radiation pressure, Phase 2 ...... 32 4.3 Integral and Monte-Carlo Test Particle methods ...... 35 4.3.1 Monte-Carlo Method ...... 35 4.3.2 Integral Method ...... 39 4.4 Summary ...... 42

5 Reference frame transformations 45 5.1 Reference frames ...... 45 5.1.1 Local ...... 45 5.1.2 Earth-fixed ...... 45 5.1.3 Inertial ...... 46 5.1.4 Orbit-fixed ...... 46 5.1.5 Body-fixed ...... 46 5.1.6 Pseudo body-fixed ...... 48 5.2 Rotation matrices ...... 48 5.2.1 Local to Earth-fixed ...... 48 5.2.2 Inertial to earth-fixed ...... 48 5.2.3 Inertial to orbit-fixed ...... 48 5.2.4 Pseudo body-fixed to body-fixed ...... 48 5.2.5 Inertial to body-fixed ...... 49 5.3 Quaternions, Euler angles and rotation matrices ...... 49

6 NRTDM data importing and (pre)processing routines 51 6.1 Data import ...... 51 6.1.1 SP3FileToOrbitFiles ...... 51 6.1.2 ImportASCIIFormatted ...... 52 6.1.3 PolynomialCalibrationParameters ...... 53 6.2 Basic conversions ...... 53 6.2.1 CalibrateAccelerometer ...... 53 6.2.2 OrbitToGeo ...... 55 6.2.3 WindCorotationSBF ...... 56 6.2.4 QuaternionToEulerAnglesConvert ...... 56 6.2.5 GOCEThrusterAccel ...... 57 6.3 Thermosphere model evaluation ...... 58 6.3.1 GeoToDens ...... 59 6.3.2 GeoToWind ...... 59 6.3.3 WindLocalToSBF ...... 60 6.3.4 WindSBFtoLocal ...... 60 6.3.5 WindVectorProject ...... 61 6.4 Force model evaluation ...... 61 6.4.1 RadiationPressureProduct ...... 61 6.4.2 ANGARA Earth radiation pressure environment model ...... 66 6.4.3 AccelerometerSubtract ...... 68 6.5 Density and wind processing ...... 69 6.5.1 DensityWindFromAccelerometerDirect ...... 69

4 6.5.2 CalibrateWithModel ...... 76 6.5.3 DensityWindFromAccelerometerIterative ...... 76 6.6 Data editing and ﬂagging ...... 79

5 6 Chapter 1 Introduction

1.1 About the project

Accelerometers carried by low-Earth orbiters such as GOCE have the ability to provide highly detailed data for improving our understanding of thermospheric density and winds. Like its predecessor missions, CHAMP and GRACE, GOCE has not been de- signed for studies of the thermosphere. Nevertheless, the application to thermosphere studies of these earlier missions has resulted in density and wind data sets containing information at unprecedented levels of precision and coverage. The algorithms for processing the data of these these earlier missions have been adapted for GOCE, and will be presented here.

1.2 Purpose of this document

The purpose of this document, according to the project’s Statement of Work, is as follows:

“Algorithm Theoretical Basis Documents (ATBD): This document shall describe into details all the algorithms, methods and models implemented for each Theme. The report shall include also all related data sources, processing steps and output data. In particular, the ATBD shall provide a complete description of all the algorithms, methods and models (both theoretical and technical) and corresponding input/output data flows, respectively. In addition, this document shall report a scientific analysis of the results driving to specific development choices and trade-offs for all the algorithms implemented for developing the whole suite of target products. Technical consid- erations justifying the selected methodologies shall be also provided.”

7 8 Chapter 2 Overall data ﬂow

2.1 Data processing ﬂow chart

Figure 2.1 shows an overview of the data processing. The processing starts with the import, pre-processing and calibration of the data. This results in four main data sources, on which the further processing is based:

• Orbit

• Attitude

• Acceleration

• Thruster actuation

The processing involves the use of thermosphere models and satellite surface force models, in combination with custom algorithms for the purpose of determining density and wind. This will be described in full detail in the remainder of this document.

2.2 Input products

The following sections describe the origin of the GOCE input products that are used in the algorithms.

2.2.1 EGG CCD (common-mode accelerations)

Common-mode acceleration measurements from the gradiometer instrument are used. The so-called sensitive pairs are used for each gradiometer axis. This data is obtained from the TU Delft HPF system, but is also readily available to GOCE users via the goce.esa.int website.

9 GOCE L1B and GOCE thruster L2 data actuation data

Pre-processing and calibration

Thruster Orbit Attitude Acceleration actuation

Radiation Empirical density Empirical wind pressure model model evaluation model evaluation evaluation

Modelled Modelled radiation composition, Modelled wind pressure temperature accelerations

Compute initial Remove non- relative velocity aerodynamic accelerations

Relative velocity Observed in SBF frame aerodynamic acceleration

Aerodynamic Density and wind model evaluation processing

Modelled density Density data Wind data

Figure 2.1 Overview of the data processing in a single ﬂow chart. Final and intermediate data products are shown in yellow boxes. Standard data processing steps are blue, the use of thermosphere models is shown in green and the use of non-gravitational force models in brown.

10 2.2.2 EGG IAQ (attitude quaternions)

Reconstructed attitude quaternion from the star cameras and gradiometer instrument are obtained from the TU Delft HPF system. This data is also readily available to GOCE users via the goce.esa.int website.

2.2.3 SST PSO (precise science orbits)

The precise science orbit ephemeris is based on GPS satellite-to-satellite tracking data. It is obtained from the TU Delft HPF system, but is also readily available to GOCE users via the goce.esa.int website.

2.2.4 AUX NOM (ion engine thrust levels)

The ion engine thrust level data is provided directly to the project team’s FTP site by ESA. The data come in XML ﬁles, which are parsed so that only a text ﬁle with a timeseries of

2.2.5 GOCE mass

The GOCE mass as a function of time is available in a ﬁle named GOCE-Mass-Properties- COG-data.GOC, available for download from the goce.esa.int website.

2.3 Space environment models

The NRTDM software incorporates several external models for the space environment. The following such models are applied in the GOCE data processing:

• NRLMSISE-00 for atmospheric composition and temperature

• HWM07 for in-track winds

• Planetary ephemerides DE405

• Solar radiation pressure environment model (eclipse, varying Sun-Earth distance) from the ANGARA phase 2 code

• Earth radiation pressure environment (Earth albedo and IR ﬂuxes at the spacecraft) model adapted from the ANGARA phase 2 code

2.3.1 NRLMSISE-00

The NRLMSISE-00 model is the latest iteration of the MSIS series of models [Hedin et al., 1977a,b, 1979, Hedin, 1983, 1987, 1991] developed at NASA’s Goddard Space Flight Cen- tre. The model development has been adopted by the Naval Research Laboratory, leading to the current version NRLMSISE-00 [Picone et al., 2002].

11 NRLMSISE-00 inputs The model takes the following inputs:

• Day of year

• Time of day

• Altitude

• Geodetic longitude

• Geodetic latitude

• Local solar time • 81-day average of F10.7, centred on the current day • Daily F10.7 for the previous day • 3-hourly ap for current time • 3-hourly ap for 3 hours before the current time • 3-hourly ap for 6 hours before the current time • 3-hourly ap for 9 hours before the current time • Average of eight 3-hourly ap indices from 12 to 33 hours before the current time • Average of eight 3-hourly ap indices from 36 to 57 hours before the current time

Outputs Outputs of the model are:

• Exospheric temperature

• Temperature at requested altitude

• Total mass density • Number densities for H, He, N, O, N2,O2, Ar and anomalous oxygen

2.3.2 HWM07 HWM07 is the latest version of the Horizontal Wind Model, which is a companion model to the MSIS series of models. This model was also originally developed at Goddard Space Flight Center [Hedin et al., 1988, 1991, 1996]. The latest version [Drob et al., 2008] was further developed at the Naval Research Laboratory and adds a separate component for storm-induced winds [Emmert et al., 2008].

12 Inputs The inputs to the HWM model are identical to those for the NRLMSISE-00 model (see above), with two exceptions. The F10.7 solar activity proxy input is not used by the model, and the code accepts only a two element array for the geomagnetic activity index ap,of which only the second element, containing the current value, is used by the model.

Outputs The outputs of the HWM07 model are: • Meridional wind (northward) • Zonal wind (eastward) This wind vector output is in the local frame (see Section 5.1), although the upwards component is always zero, and the north- and eastward axes are switched in the model subroutines arguments, with respect to the reference frame deﬁnition.

2.3.3 Planetary ephemerides JPL DE405 The JPL planetary ephemerides DE405 [Standish, 1998] are implemented in NRTDM for the computation of the position of the Sun with respect to the Earth and the satellite. This is necessary in the calculation of radiation pressure accelerations.

Inputs The only input to the ephemerides calculation is the time.

Outputs As output, the position of the Sun with respect to the Earth in the inertial frame is re- turned.

2.4 Auxiliary input data

The NRTDM software requires two types of auxiliary data: • Solar and geomagnetic ﬂux data used in calling the thermosphere density, temperature and wind models (see Section 2.3). • Earth orientation parameters for the conversion between inertial and Earth-ﬁxed reference frames (see Section 5.1).

2.4.1 Solar and geomagnetic activity data

The NRLMSIS and HWM models use the F10.7 solar ﬂux proxy and ap geomagnetic activity indices, to represent variations in the energy input from the Sun into the atmosphere. The data are downloaded from NOAA’s National Geophysical Data Centre server, at the following URL: • ftp://ftp.ngdc.noaa.gov/STP/GEOMAGNETIC DATA/INDICES/KP AP • ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SOLAR RADIO/FLUX

13 2.4.2 Earth orientation parameters The conversion between inertial and Earth-ﬁxed coordinates (see Chapter 5) requires the use of Earth-orientation parameters. These are downloaded from the following location:

• http://hpiers.obspm.fr/iers/eop/eopc04/eopc04 IAU2000.62-now

2.5 Output products

The main outputs of the algorithm, to be provided as a function of time, are

• Density (kg/m3)

• Crosswind (m/s), vector with east, north, up components

Additional fields from the NRTDM data storage can be provided alongside this data. The most important candidate data fields for inclusion in the final product are:

• Geodetic latitude, longitude, height, local solar time

• Quality ﬂags

• Modelled density

• Modelled wind

• Modelled wind projected on crosswind direction

14 Chapter 3 Data calibration

3.1 Accelerometer and ion engine calibration using GPS data

Ideally, the non-gravitational accelerations ang experienced by GOCE are equal to the so-called common-mode accelerations acm observed by its gradiometer plus the thrust T from the ion engines divided by the mass m of the satellite:

= + ang acm T/m (3.1)

Equation 3.1 indicates that (1) the common-mode accelerations acm and (2) the thrust vector T need to be properly calibrated. Ad (1). The common-mode accelerations are obtained by taking the average of observations by pairs of accelerometers. In fact, the three most sensitive combinations are taken, one for each direction of the gradiometer reference frame (GRF), which is pair 1&2 for the X-axis, pair 3&4 for the Y-axis, and pair 5&6 for the Z-axis in the GRF. The center of the gradiometer almost perfectly coincides with the satellite’s center of mass. By taking these common-mode combinations, rotational terms are almost perfectly eliminated, provided the accelerometers are properly calibrated and aligned. Based on the precision requirements for the GOCE satellite gravity gradient (SGG) observations, it is assumed that the scale factors for the accelerometers are known very well and that in fact these scale factors can be considered to be equal to ONE (1.00) for the Level 1b common-mode accelerations (”EGG CCD” data, see HPF [2010]). Also, the requirements for the alignment of the accelerometers are very stringent. However, the requirements for the SGG observations are valid for the measurement bandwidth (MB) of 0.005-0.1 Hz, meaning that the accelerometer observations can and in fact are offset by (drifting) biases. These biases need to be known before the common-mode accelerometer observations can be used for thermospheric density and winds determination. These biases can be estimated by precise orbit determination (POD) by the same methodology applied for calibrating e.g. the CHAMP and GRACE accelerometers Bruinsma and Bian- cale [2003], Bruinsma et al. [2004], Visser and van den IJssel [2003], where the common- mode accelerations represent the non-gravitational accelerations in the GOCE dynamic

15 force modeling for the POD. Moreover, the POD provides a validation of the assumption that the scale factors are equal or close to ONE as well. For GOCE, this methodology has been implemented and is used operationally in the framework of the GOCE High-level Processing Facility (HPF). The architecture of this implementation is described in HPF [2007] (pp. 51-53) and Bouman et al. [2011], for which the GOCE standards were adopted Gruber et al. [2010]. Please note that time series of kinematic orbit positions are used as the observation type. These kinematic position estimates can be considered a condensed set of GPS Satellite-to-Satellite Tracking (SST) observations. For this project, a few enhancements were made to the HPF implementation. These enhancements are:

• Use of GOCO02S gravity ﬁeld Goiginger et al. [2011] model instead of EIGEN-5S Gruber et al. [2010]: the post-launch GOCO02S model provides a better representation of the gravitational forces acting on GOCE than the pre-launch EIGEN-5S model leading to smaller aliasing of gravitational errors into estimated common- mode acceleration calibration parameters;

• Use of the kinematic Precise Science Orbit (PSO) instead of the Rapid Science Orbit (RSO). The claimed accuracy of the PSO product is of the order of a few cm in terms of position Bock et al. [2011]. The PSO solutions have a length of 30 hr with a 6-hr overlap between consecutive days. For the estimation of calibration parameters, the full PSOs are used, i.e. an arc length of 30 hr is adopted;

• Use of star tracker mounting matrices from the GOCE gradiometer calibration (AUX EGG DB): the star tracker observations are used instead of the combined quaternion product EGG IAQ HPF [2010] because of better continuity (less data gaps): the accuracy of the orientation of the GRF is crucial for estimating calibration parameters for the common-mode accelerations.

Because of the quality of the gravitational force modeling, it was found that 3-dimensional kinematic orbit fits of around 15 cm could be obtained by estimating only the begin position and velocity plus three common-mode acceleration biases for every 30-hr arc (fixing the scale factors to ONE). Thus only 9 parameters were estimated for each arc. Figure 3.1 displays the estimated biases for a period covering November 2009 - Septem- ber 2011. It can be observed that the estimates for the X axis (predominantly the flight direction) are very stable with an Root-Mean-Square (RMS) of fit of 0.15 nm/s2 to a linear model representing a bias of -187.5 nm/s2 (at start time 1 November 2009) and a drift of 0.0010 nm/s2/day. For the Y-axis (predominantly cross-track) the RMS value is equal to 12 nm/s2 for a bias value of -310 nm/s2 and a drift of 0.14 nm/s2/day. For the Z-axis (predominantly radial direction), the results are less reliable with an RMS of 66 nm/s2,a bias of 97.8 nm/s2 and a drift of -0.02 nm/s2/day. Ad (2). It is assumed that the thrust magnitude and direction of the ion engines are well known and accurately calibrated. However, a calibration of the ion engine thrust magnitude can be done (to some extent) as well by POD considering the fact that regu- larly maneuvers are carried out, where the GOCE ion engines have different thrust levels. To this aim, the same implementation as for the estimation of the common-mode acceleration biases can be used with the following changes:

• The common-mode accelerations are no longer used;

16 Figure 3.1 Bias estimates for sensitive common-mode combination, i.e. accelerometer pair 14 for the X-, pair 25 for the Y-, and pair 36 for the Z-axis.

17 • Piecewise linear empirical accelerations are deﬁned with time interval of 15 min between the notes for the X, Y and Z axes of the GRF. These accelerations are estimated and represent the GOCE non-gravitational accelerations.

Please note that the empirical accelerations are defined in the GRF allowing a direct comparison with the observed common-mode accelerations. For a 30-hr arc, the total number of estimated accelerations is equal to (30 × 4 + 1) ∗ 3 = 363. With the begin position and velocity, the total number of estimated parameters is then equal to 369. The above approach has been successfully implemented for other missions van den IJssel and Visser [2005, 2007] and has been tested for a few selected GOCE days, see Figures 3.2-3.4. Jumps in the non-gravitational acceleration level can especially be observed for the X- axis. For 15 June 2009 (during commissioning phase) and 7 March 2010, these jumps are relatively large: of the order of 1700 and 950 nm/s2, respectively. At the end of 14 August 2011, the jump is relatively small, about 400 nm/s2. It is interesting to see that similar jumps can be observed for the Z axis, but at a much smaller level. The differences between the provided ion engine thrust levels can be compared with these jumps. A similar comparison can also be done with the common-mode accelerations, provided that these are properly calibrated. Time series of accelerations derived from the ion engine thrust values, the common-mode accelerometer observations and those estimated by POD are displayed together in Figure 3.5 for the X-axis for a few hours before and after the change of ion engine thrust level. Especially for the selected day during the commissioning phase (15 June 2009), the jump is clearly visible in all time series. As the GOCE mission progresses, so does the atmospheric drag due to increased levels of solar activity. The ion engines nominally compensate the atmospheric drag and the ion engine thrust values thus display the larger variations of the atmospheric drag for the later stages of the GOCE mission. This can be observed clearly in the middle and bottom plots of Figure 3.5. These increased levels of atmospheric drag fluctuations make it more difficult to clearly identify jumps, and thus match different ion engine thrust levels with different levels of the estimated non- gravitational accelerations or common mode observations. Since the most dominant aerodynamic drag fluctuations occur at a frequency of 1 cycle per orbit revolution, a comparison has been made between the averaged values of the different time series for periods equal to one orbital revolution before and after each maneuver. A margin of 5 min before and after the maneuver was used in order to eliminate the transition period of ion engine thrust levels. The orbital period was taken − equal to 5400 s. Thus, for each maneuver time tman the period before covers tman 5700 − − − to tman 300 s and the period after covers tman 300 to tman 5700 s. The averaged acceleration levels for the different time series are displayed in Table 3.1 for the selected maneuvers. For the first two selected days, 15 June 2009 and 7 March 2010, the consis- tency of the jumps in the mean values (red and blue) is of the order of a few percent: respectively 1690.8, 1677.3, 1696.7 nm/s2 and 960.1 , 985.3, 977.7 nm/s2. For the latest selected day, 14 August 2011, this jump derived from the ion engine thrust values is equal to 423.5 nm/s2 compared to 352.0 and 340.9 nm/s2 for the jumps derived from the POD and common-mode values (green), a discrepancy of up to 25%. The ion engine thrust values are available with a time interval of 10 s compared to 1 s for the common-mode accelerations and 15 min for the values estimated by POD. Com- paring with values determined by POD thus requires an averaging interval of at least 15 min, during which the aerodynamic drag variations would not average out. Using an

18 Figure 3.2 Non-gravitational accelerations: common-mode vs. estimated by POD. Time is relative to 14 June 2009 00:00 (note the difference in scale).

19 Figure 3.3 Non-gravitational accelerations: common-mode vs. estimated by POD. Time is relative 6 March 2010 00:00 (note the difference in scale).

20 Figure 3.4 Non-gravitational accelerations: common-mode vs. estimated by POD. Time is relative to 14 August 2011 00:00 (note the difference in scale).

21 15 June 2009 10:16:35

7 March 2010 14:00:33

14 August 2011 23:58:33

Figure 3.5 Non-gravitational accelerations for the X-axis: common-mode, estimated by POD and derived from the ion engine thrust values. The approximate maneuver time is indicated above the plots.

22 Table 3.1 Averaged acceleration values for the orbital revolution before and after the selected maneuvers (values in nm/s2).

Before After Difference Date Ion engine thrust 15 June 2009 834.9 2526.8 1690.8 7 March 2010 2397.1 3355.5 960.1 14 August 2011 2600.8 3025.8 423.5 Precise Orbit Determination 15 June 2009 -60.5 1616.8 1677.3 7 March 2010 -2.2 983.1 985.3 14 August 2011 -0.7 351.3 352.0 Common mode 15 June 2009 108.0 1804.7 1696.7 7 March 2010 186.4 1164.1 977.7 14 August 2011 188.5 529.4 340.9

averaging interval of 1 orbital revolution only partially takes care of this. Of course, the quality of the averaging depends on the evolution of the orbit and of atmospheric conditions, which is the reason of the larger discrepancy found for the selected day later on in the GOCE mission. A comparison with common-mode accelerations would be more straightforward, provided the ion engine thrust values would be known with a smaller time interval as well. Figure 3.6 provides a zoom-in of the associated acceleration values close to the maneuver. The jumps caused by a change of thrust level can be clearly observed, but again a comparison between ion engine thrust values and common-mode accelerations is not straightforward for 14 August 2011 with the given sampling rates.

23 15 June 2009 10:16:35

7 March 2010 14:00:33

14 August 2011 23:58:33

Figure 3.6 Zoom-in of non-gravitational accelerations for the X-axis: common-mode and derived from the ion engine thrust values. The approximate maneuver time is indicated above the plots.

24 Chapter 4 Non-gravitational force modeling

4.1 GOCE satellite geometry model

4.1.1 Modelling approach There are different possibilities to model a spacecraft for surface force calculations. Their complexity ranges from spheres or simple rectangular shapes up to a detailed overall modelling of the local geometry. It depends on the application which degree of complexity is best suited. It is obvious, for example, that it will not be possible to model lift with spheres, or torques with symmetrical polyhedrons. For the present contract it was decided to use a detailed modelling of the geometry, for several reasons:

• A detailed model is most capable of determining also detailed force and torque coefﬁcients in all coordinate directions.

• Using appropriate methods for integrating the local coefﬁcients also mutual shadowing of different parts of the geometry can be taken into account quite generally, without the need of a case-by-case analysis as for simpliﬁed panel methods, for example.

• The availability of a software for non-gravitational force calculations (ANGARA) which uses such a detailed geometry modelling.

In ANGARA [Fritsche et al., 1998] the geometric modeling of the surfaces of a spacecraft is based on the principle of element-by-element deﬁnition. In Fig. 4.1 the structure of the geometric modeling is given. Each spacecraft is represented as a combination of separate elements. The initial geometric elements are called geometric primitives. This set of geometric primitives includes ﬁrst-order elements such as rectangle, polygon, circle, and ellipse; second-order elements such as cylinder, cone, sphere, ellipsoid, and paraboloid; and also sets of cross sectional

25 Geometric model

GM1.1

GM1.2

GM1.3 Compound Level

C4 C5

C2 C1 C3

Primitive Level

Cone Sphere Rectangle Triangle Figure 4.1 Structure of the geometric modelling.

26 Y

L R2

R1 X Phi1

Z Phi2

Figure 4.2 Example of a primitive shape and its parameters

elements whose surfaces are determined by points in parallel sections. Each primitive has its own parameters, which define the shape and size of the primitive. On the second level, a combination of primitives forms a compound. For compounds, a set of parameters can be declared, which influence the compound shape and size. This process is named parameterization. Compounds can be also used in other compounds. A compound with a fixed set of parameters transforms into a Geometric model. From one compound several models can be made, where each of them has its own fixed set of parameters (like solar array rotation angle). Geometric primitives are defined by several geometric parameters in its own local coordinate system. For each kind of primitive a special subroutine exists which can provide the transformation from the analytical sur-

Parameter Range Explanation R1 0 < R1 < ∞ Radius in the ﬁrst cross section R2 R1 < R2 < ∞ Radius in the second cross section L 0 < L < ∞ Length of the cone Phi1 0 < Phi1 < 360 Angle (in degrees) between the OY axis and the segment origin in the OYZ plane Phi2 0 < Phi2 < 360 Angle (in degrees) of the segment in the OYZ plane. step1 0 < step1 Partition step size along the OX axis step2 0 < step2 Partition step size along the OY axis normal ” + ”or” − ” Normal sign (inner/outer surface = +/−)

Table 4.1 List of the primitive parameters

27 Figure 4.3 ANGARA GOCE geometry

Figure 4.4 ANGARA GOCE panelized geometry

face description to the panelized description. An example of a primitive deﬁnition is shown on Figure 4.2 and Table 4.1. The surface of each geometric element is split into triangular panels approximately equal in size. The sizes of partition steps in two quasiorthogonal directions are deﬁned as element parameters. Thus, a panel model is composed of plane triangular panels approximating the surfaces of all the elements included in this object.

4.1.2 Modelling GOCE The GOCE geometry model was created from CAD drawings at a high level of detail. This level of detail in the geometry should translate into higher accuracy in the force coefﬁcients, and consequently in the density and wind results. In fact, at the current level of detail of geometric modelling, it is likely that other aspects, such as uncertainties in the gas-surface interaction, dominate the force coefﬁcient error. Figure 4.3 shows the GOCE geometry modelled with ANGARA. The surface consists

28 of 209 primitives. Figure 4.4 shows the ANGARA panel model of GOCE. The number of panels is 5402.

4.1.3 Reference frames There are three different reference frames to consider when a satellite is modelled for the calculation of non-gravitational forces and moments:

1. Geometric reference frame. This is the coordinate system in which the satellite geometry is modelled. In principle its coordinate directions are arbitrary, but its relation to the body-ﬁxed axes (see below) has to be known. In ANGARA this relation is ﬁxed.

2. Body-fixed reference frame. The geometric reference frame (see above) is body- fixed as well, but the so-called Spacecraft Body-Fixed (SBF) frame has a fixed relation to the aerodynamic reference frame (see below). In ANGARA the relation between the axis of the geometric (index g) and body-fixed system (index b) are: • = − xg xb • = − yg zb • = − zg yb The spacecraft body-fixed frame for GOCE, and its relation to other frames is further described in Section 5.1.5. Figure 5.1 shows the body-fixed coordinates for GOCE.

3. Aerodynamic reference frame. This frame is actually determined only by the inverse wind vector. The x-axis of this coordinate frame is given by the opposite direction of the free-stream direction (wind) relative to the satellite. Therefore the aerodynamic attitude of a satellite is determined by only two angles instead of a full set of three e.g. Euler angles. The attitude is undetermined relative to a rotation about the wind vector. The two angles determining the attitude of the satellite relative to the wind vector (or more exactly: the direction of the wind vector in the spacecraft body-ﬁxed frame) are the angle of attack α and the side-slip angle β. The ( ) aerodynamic angles can be computed from the directions Vx, Vy, Vz of the inverse wind vector by: • = α β Vx cos cos • = β Vy sin • = α β Vz sin cos

4.2 ANGARA two-phase approach

The traditional approach to address the problem of modeling non-gravitational effects is to consider a simpliﬁed geometry of a given spacecraft on a given orbit and to compute the force and torque acting on the S/C, considering the ﬂuxes of photons and molecules as function of the orbit phase. Disadvantages of this approach are the restricted representation of the real geometry and a time-consuming orbit propagation.

29 Figure 4.5 Body-ﬁxed coordinate system of GOCE

In the approach used in ANGARA, the process of computing perturbing forces and torques acting on a specific spacecraft has been organised in two phases. In Phase 1, the spacecraft model is established, and the coefficients of force and torque for aerodynamic and radiative actions are computed in advance for the expected range of flow/radiation attack angles and speed ratios/wavelengths by means of test particle or integral methods. The calculation results are stored as matrix coefficients in a database. The contents of the database can be distributed as data file. In Phase 2, the aerodynamic and radiative environment of the spacecraft are established for the actual state vectodr and attitude of the satellite at a given epoch. The resulting forces and moments are computed by interpolating the coefficients in the data matrix computed in Phase 1 for reference conditions, and by multiplication of these coefficients with corresponding normalisation factors derived from the actual enviromental conditions. In this way, the major part of the whole work is concentrated in Phase 1. In Phase 2, the results of Phase 1 only have to be adjusted for user-defined environmental conditions. The effort for force and torque calculation becomes independent of the complexity of the spacecraft geometry, which allows to calculate aerodynamic and radiation forces on complex spacecraft in near real-time.

4.2.1 Aerodynamic analysis, Phase 1 For the free-molecular flow each gas species can be considered independently, and aerodynamic coefficients of forces and moments depend only on the free stream molecular speed ratio S∞ and wall temperature TW of a spacecraft for fixed gas/surface interaction properties. If a similar law of gas/surface interaction is used for all gas species, it is only necessary to perform the aerodynamic computations for different S∞i, covering the whole range of free-stream velocities U∞, molecular masses mi, and ambient temperatures T∞.

= U∞ S∞i · · ∞ 2 k/mi T i Then, the following procedure can be used for Phase 1:

30 1. Determination of the range of all possible values S∞ for all atmospheric species;

2. Calculation of aerodynamic coefficients for a set of S∞i and for a specified range α β re f of angles of attack , side slip angles , and reference wall temperatures Tw of a spacecraft. The result of the computation is a matrix of coefficients; C 3. Storage of the following aerodynamic coefficients of force ( a) in the results database: Cinc Cout + Cmr a , a a

where ‘inc’, ‘out’ and ‘mr’ stands for incident, reﬂected and multiply reﬂected.

4.2.2 Aerodynamic analysis, Phase 2

1. Determination of S∞i for user defined free-stream velocity U∞, temperature T∞ and atmospheric composition ni (number density of gas species ’i’); 2. Interpolation of aerodynamic coefficients of forces and moments for all values of α β S∞i and for a user defined spacecraft orientation , ; Cout + Cmr 3. Correction of a a for a user defined wall temperature Tw of the spacecraft Cout + Cmr = Cout + Cmr · re f a a a a re f Tw/Tw (4.1) Tw Tw (4.2)

4. Computation of the aerodynamic force coefﬁcient from all contributions for the gas species.

C = Cinc + Cout + Cmr a,i a a a (4.3)

5. Determination of the effective aerodynamic force coefﬁcient vector, by summation over the atmospheric constituents, weighted by their partial contribution to the total mass density. N ρ C = i C a ∑ ρ a,i (4.4) i=1

6. Determination of aerodynamic forces 1 2 inc out mr F = ρU∞ · A · C + C + C (4.5) a 2 ref a a a

4.2.3 Radiation pressure analysis, Phase 1 The calculations are conducted similarly to the calculations of aerodynamic coefﬁcients for Phase 1.

1. Calculation of the radiation coefficients of force C in the body-fixed coordinate system for a specified range of α, β (spacecraft view angles towards the radiation source), for molecular speed ratio S = ∞, for the visible spectrum, and for the IR spectrum.

31 y x

z g

North pole Z A a i i z r i R s R 0i yg cos ( a ) > 0 O Y i if the element of Earth surface is visible from the spacecraft

x g Figure 4.6 Integration of indirect radiation over the Earth surface

2. Calculation of the self-radiation coefﬁcients of force for the illuminated part of the sh spacecraft (Csel f ) and for the shadowed part (Csel f ) at reference wall temperature re f α β Tw and given direction towards the sun and . 3. Storage of the radiation coefﬁcients in the results database:

sh sh C, M, Csel f , Msel f , Csel f , Msel f

4.2.4 Direct radiation pressure, Phase 2 In Phase 2, direct radiation pressure results can be retrieved by the following scheme:

1.1 Deﬁnition of the attitude and position of a spacecraft at a given epoch.

1.2 Determination of the effective, seasonal dependent radiation ﬂux level.

1.3 Determination of α and β (spacecraft view angles towards the sun)

2. Interpolation of radiation coefﬁcients of forces and moments from reference results for given α, β.

3. Correction of radiation coefﬁcients for given spacecraft temperature Tw. 4. Determination of forces and moments at effective direct solar radiation ﬂux level for the visible and IR spectrum.

4.2.5 Indirect radiation pressure, Phase 2 Principles of the calculation The Earth surface is assumed to be divided into elementary cells bounded by meridians and small circles of latitude in a standard Mercator projection. Its reﬂective and radiative

32 properties are assumed constant within a cell but may change from one cell to another. When calculating the radiation fluxes at the spacecraft location, one has to take into account only those cells of the Earth surface that are seen from the spacecraft. A cell shall be assumed visible, and it shall be taken into account for the flux integration, if its center is seen from the spacecraft. The relative position of the spacecraft and the cell is character- r = R − R ized by the vector i s 0i (See figure 4.6). An obvious condition of visibility is that γ r R π (γ ) > i, the angle between the vectors i and 0i, is smaller than /2, and hence cos i 0. r R This means that the scalar product of vectors i and 0i should be positive: r · R > (R − R ) · R > i 0i 0, or s 0i 0i 0.

Each cell of the Earth surface is a source of infrared radiation and reflected visible sun light (albedo). Let the intensity of the infrared radiation emitted from the unit cell area in (IR) (VL) all directions be Ii , and the corresponding value for the reflected visible light be Ii . The angular distribution of this radiation is in line with the Lambert law for a diffusely reflecting body: the radiation intensity is proportional to the cosine of the angle γ counted from the normal to the body surface: I(γ)=I(0) cos γ where (as it can be easily verified by integration over the half space) I(0) equals the total intensity of radiation emitted in all directions divided by π. Taking into account that electromagnetic radiation intensity decreases with distance according to the inverse-square law, the following values are obtained for radiation fluxes per unit area at the spacecraft location:

(IR) γ (VL) γ ( ) I Ai cos i ( ) I Ai cos i G IR = i , G VL = i (4.6) i π 2 i π 2 ri ri In contrast with the infrared contributions, an earth surface element can only emit radiation in the visible spectrum if it is in line of sight of both the satellite and the sun.

Calculation scheme 1.1 Deﬁnition of the attitude and position of a spacecraft at a given epoch relative to the Earth-ﬁxed coordinate system.

1.2 Adaptation of the re-radiation characteristics of the grid of the Earth surface elements for seasonal effects and solar aspect angle.

1.3 Determination of the satellite related incident angles α and β (angle of attack and side slip angle) for each surface element of the visible Earth disk.

2. Determination of the radiation pressure contribution of each element by interpolation and ﬂux level adjustment of the reference result.

3. Calculation of the effective radiation pressure by summarizing over all surface elements of the visible Earth disk (for earth IR re-radiation), and of the visible, illuminated Earth disk (for earth albedo re-radiation).

Figure 4.7 illustrates the software modules and the data flow in Phase 1. Output data to Phase 2 are the albedo and IR maps, and the calculated coefficients. The implementation of Phase 2 is user-specific.

33 Manager DB

Geometry

Modules Model

Job1

Albedo PreView

To Phase 2 maps matrix Figure 4.7 Calling interfaces and data ﬂow in Phase 1.

34 4.3 Integral and Monte-Carlo Test Particle methods

In Phase 1, the coefﬁcients of force and torque due to aerodynamic and radiation actions on the modeled spacecraft are calculated. This can be done with two different methods:

1. Test Particle Monte Carlo method (TPMC)

2. Integral Method

The main advantage of the Test Particle Monte Carlo method is that the effect on the acceleration of multiple reﬂections of the gas atoms can be taken into account. In reality for GOCE, such multiple reﬂections will occur at the intersections of the wings and body, which form concave corners. For this reason, the results of this method will be used in the data processing. The main disadvantage of the Monte-Carlo method is the random error due to the use of a limited number of test particles. This error is reduced when the number of test particles is increased, but at the cost of computation time. The integral method, on the other hand, gives a smooth output. This method has been used for validation of the Monte-Carlo method.

4.3.1 Monte-Carlo Method This method takes into account the ﬁnite Mach number, multiple reﬂections, and shadowing effects.

General scheme of the Monte-Carlo calculation The Test Particle Monte-Carlo method simulates flows by tracing the trajectories of test particles in a computational domain which contains the spacecraft. It is assumed that each particle represents a large amount of real molecules. During its motion a particle can strike the surface of the spacecraft and be reflected according to a given gas-surface interaction model. When striking the surface, the particles exchange a part of their momentum and energy with the moving spacecraft. The reflected particles either can strike the body again or leave the computational domain (see figure 4.8). After tracing a large amount of particles, the integral aerodynamic (or radiation) characteristics of the spacecraft (drag, lift, moments, etc.) are calculated. In the following sections the single steps of the calculation procedure are explained.

Start of a particle trajectory Aerodynamic case

• Each test particle starts from one of the sides of the computational domain with the = ∑6 probability pl Ql/ i=1 Qi, where Ql is the particle number ﬂux across the side n number l with area Fl and normal direction l: = (ξ · n ) (ξ) ξ · (ξ)=π−3/2 −3 − (ξ − S )2 Ql l f d Fl f c exp ∞ (ξ·n )> l 0 (4.7)

35 Figure 4.8 General scheme of Monte-Carlo calculation

where c =(2RT)1/2 is the most probable thermal speed, S∞ = U∞/c is the free- stream speed ratio, c is the thermal velocity vector, U = U∞ + c, ξ = U/c . The start points are uniformly distributed over the faces.

• The velocity vector ξ of the particle is selected. The density of distribution of the velocity component directed along the face normal is

2 · ξ exp{−(ξ − S )2} p(ξ )= i √ i i i {− 2} + π ( + { }) exp Si Si 1 erf Si and for tangential directions it is

(ξ )= 1 {−(ξ − )2} p j π exp j Sj

(ξ )= 1 {−(ξ − )2} p k π exp k Sk

Radiation case

• there is no thermal scattering for light

• all light beams have the same velocity equal to the speed of light c, it can be included into the normalization factor, and the velocity of all modeled particles can be assumed equal to 1.

• the probability of a photon to enter into a certain face of the computational domain can be derived from (4.7), which for one single velocity reduces to =(ξ · n ) · (ξ · n ) > Ql l Fl; for l 0

36 • the normalised photon velocity components along the coordinate axes are

ξ = (α) · (β) i cos cos

ξ = (α) · (β) j sin cos ξ = (β) k sin where α and β are the radiation angles of attack and side slip.

Search for the intersection point

For the detection of the intersection point of a test particle with the spacecraft surface the following features are used:

• the spacecraft surface is described by a set of triangular panels;

• the computational domain is split into a set of cells of equal size;

• for each cell, the corresponding panels are determined;

• to check for intersection points, the cells through which the trajectory passes, are determined by a fast browsing algorithm;

• only the panels located in these cells are checked for intersection.

Aerodynamic reﬂection

The particle reflection of a molecule on the satellite surface is calculated by the Maxwell gas-surface interaction model. It is constructed on the assumption that a fraction 1 − σ of particles is specularly reflected from the surface, and a fraction σ is re-emitted diffusely, with the Maxwellian distribution: 2 √ − −3 ξr (r ξr)=( − σ) (r ξr − (ξr · n)n)+σπ 3/2 − = fr , 1 fi , 2 c r exp 2 cr 2RTw c r ξ n r where r is the reflected velocity, and is the external normal of the surface at point . The velocity of diffusely reflected particles is sampled in a local polar coordinate system, directed along the outer normal to the surface of the vehicle: | ξ |= − ( ) r cr ln R1R2 ; (ψ)= cos R3; φ = π 2 R4.

where | ξr | is the modulus of velocity of reﬂected particles; ψ, φ are elevation and azimuth angles; R1, R2, R3, R4 are pseudo-random numbers.

37 Radiation A photon interacting with the satellite surface can have different behaviours:

• it can be absorbed by the surface; then it will be converted into heat or electricity (solar panel); • it can be specularly reﬂected (as by a mirror); • it can be diffusely reﬂected.

The reﬂection is computed by using the following steps:

1. the particle is absorbed with a probability α. If it is absorbed, then the particle is not traced further, and a new particle is launched. = ρ (ρ + ρ ) 2. specular reﬂection occurs with a probability P S/ S D = ρ (ρ + ρ ) 3. diffuse reﬂection takes place with a probability P D/ S D . The Monte-Carlo technique selects one of these possibilities, based on their associated probabilities of occurence. This corresponds to α + ρ + ρ S D

Multiple reflections and termination of a trajectory If a particle is reflected, the next possible point of intersection with the spacecraft surface is calculated. In case of a hit, the total incident momentum is adjusted by adding the momentum of this secondary reflection. If the trajectory of the reflected particle intersects the boundary of the computational domain, the tracing is then finished, and the start of a new particle is simulated. The tracing of the particle is terminated when the number of particle collisions from the surface reaches a given number of hits.

Calculation of normalised aerodynamic characteristics Interaction properties such as momentum, angular momentum and energy received from each test particle at collision and reﬂection from the spacecraft surface are collected in corresponding counters. On the basis of the collected information the aerodynamic coef- ﬁcients are calculated in the following manner:

N N = ξ j = (x · ξj ) Ck A ∑ k; Mk A ∑ k /Lref j=1 j=1 The factor A is used for normalization. Its value for the aerodynamic analysis is

6 = ( 2 ) A 2 ∑ Ql / S∞ NAref l=1 and for the radiation analysis it is

6 = ( ) A 2 ∑ Ql / NAref l=1

38 Estimation of the statistical errors To determine the statistical error, the calculation is performed batch-wise. All trajectories are divided into groups (batches), each consisting of Np trajectories. For each batch, the characteristics can be calculated by

Np η = ζ 1/Np ∑ i i=1 where ζ is a random variable. The mean value is

Kp η = η ¯ 1/Kp ∑ k; k=1 The dispersion is Kp 2 = (η − η)2 Sd 1/Kp ∑ k ¯ ; k=1 The ﬁnal absolute statistical error of calculation may be written as | ζ N − | < 2 ( − ) I tKp−1,1−ε Sd / Kp 1

4.3.2 Integral Method With the Integral Method the aerodynamic and radiation coefficients can be computed in an alternative way. All effects considered in the Monte Carlo method are included, except for multiple reflections. have an influence on the aerodynamic force computation,

Overview In the integral method the velocity distribution functions of the incident and the reflected particles, which are used in the Monte Carlo method as probability functions for each launched particle, are integrated to get an expression for the local stress σ. The aerodynamic (radiation) stress is equal to the momentum flux transferred from the gas molecules (light photons) to the surface. This momentum exchange depends on the flux of incident particles and on the gas(photon)-surface interaction law. Forces and torques acting on the spacecraft are calculated by integrals over the spacecraft surface S exposed to the flow/radiation: F = τ(r) dS (4.8) S M = (r − r ) × τ(r) cm dS (4.9) S τ(r) r where is the local stress and cm is a moment reference point. By appropriate separation of geometrical reference quantities (area Aref, length Lref) the integrals can be expressed in non-dimensional form C = 1 c (r) F τ dS (4.10) Aref S C = 1 (r − r ) × c (r) M cm τ dS (4.11) Aref Lref S

39 The relation between forces (moments) and their corresponding coefﬁcients depends on the physical nature of the ﬂow.

Aerodynamic case 2 In the aerodynamic case τ = q∞cτ, where q∞ = ρ∞U∞/2 denotes the dynamic pressure of the free stream, and F = C A q∞ Aref F (4.12) M = C A q∞ Aref Lref M (4.13) The local stress depends on the gas-surface interaction. Using the Maxwell model, where it is assumed that a fraction σ of the molecules impinging on the spacecraft surface is reflected diffusely and the remaining fraction (1 − σ) is reflected specularly, one gets C = 1 (σ c (r)+( − σ) c (r)) F τ,di f f 1 τ,spec dS (4.14) Aref S C = 1 (r − r ) × (σ c (r)+( − σ) c (r)) M cm τ,di f f 1 τ,spec dS (4.15) Aref Lref S with Π( ) χ( ) χ( ) c = √1 Sn + 1 TW Sn n + √1 St Sn t τ,di f f (4.16) π S∞2 2 T∞ S∞2 π S∞2 Π( ) c = √2 Sn n τ,spec (4.17) π S∞2 where √ Π( )= (− 2 )+ π( 2 + 1)( + ( )) Sn Sn exp Sn Sn 1 erf Sn (4.18) √ 2 χ(S )=exp(−S2 )+ πS (1 + erf (S )) (4.19) n n n n = 2 with speed ratio S∞ mU∞/2kT∞ and its components Sn, St normal and tangential to the surface. The local stress coefficient depends on the wall temperature. In Phase 1 of the two- phase approach the coefficients are calculated for a fixed wall temperature Tre f . In Phase 2 the wall temperature TW can be arbitrary and the coefficients have to be corrected ac- cordingly. This is achieved by splitting the stress coefficient in a different way: c = σ c Θ TW +(− σ) c ( Θ) τ τ,di f f S∞, , 1 τ,spec S∞, (4.20) T∞ 2 − σ Π(S ) σ S χ(S ) T σ χ(S ) = √ n n + √ t n t + W n n (4.21) π 2 π 2 2 S∞ S∞ T∞ 2 S∞ (0) TW (1) = cτ (S∞, Θ, σ)+ cτ (S∞, Θ, σ) (4.22) T∞ The spacecraft’s coefficient C , for example, then becomes F (0) TW (1) ( )= ( ∞ α β σ)+ ( ∞ α β σ) CF TW CF S , , , CF S , , , (4.23) Tre f

(0) (1) (0) (1) In Phase 1, CF , CF , CM and CM are computed and stored in the database. The temperature correction is applied in Phase 2.

40 Radiation case

In the radiation case τ =(j∞/2c) cτ, where j∞/c denotes the momentum ﬂux of the incoming radiation from the radiation source (direct or indirect), and

j∞ F = A C (4.24) R 2c ref F j∞ M = A L C (4.25) R 2c ref ref M The normalisation with half of the momentum flux was chosen in analogy to the aerodynamic case, where the normalising dynamic pressure q∞ corresponds to one half of the momentum flux of the molecule flow. The local stress depends on the photon-surface interaction. Using the photon-surface interaction law introduced in the Monte Carlo section, we have to consider three interaction types: absorption, specular reflection, and diffuse reflection. The sum of these contributions is given by c = α c + ρ c + ρ c τ/2 τ,A S τ,S D τ,D (4.26) = α u + ρ (un) n + ρ (u − n ) 2 S D 2 /3 (4.27) 2 =(un) (1 − ρ )u +(2ρ (un) − ρ )n (4.28) S S 3 D α + ρ + ρ = α u where the condition S D 1 was used to eliminate . is the direction of the incoming photons. In a body-fixed reference frame it can be expressed by the angle of attack and side slip angle

u = −(cos α cos β, sin β, sin α cos β) (4.29)

The integral coefficients CF, CM are functions of the attack angles and the reflection coefficients only, therefore no special treatment for Phase 2 is required here. The satellite surface itself emits thermal radiation, which exerts a recoil momentum on the spacecraft body. The energy emitted by a surface element per unit time is given by Stefan-Boltzmann’s law ( )=σ 4 jth TW BTW Assuming the spacecraft surface to radiate according to Lambert’s cosine law, the recoil momentum due to the thermal radiation is given by

τ = −2 jth n 3 c A local stress coefﬁcient can be deﬁned by normalising with the corresponding recoil τ ( )=( )σ 4 momentum for a black body at a reference temperature, 0 Tre f 2/3c B Tre f : 1 τ = τ (T ) cτ 2 0 re f where again a factor 1/2 has been added (see above). Then, simply 4 TW cτ = −2 n Tre f

41 Now, integration over the spacecraft surface to get the integral coefficients CF and CM will yield always zero, if the wall temperature (and emission coefficient ) are assumed to be constant (like in the aerodynamic case). In ANGARA, at least two surface temperatures are used in the radiation analysis, assuming that parts of the surface which are illuminated by incoming radiation have a different temperature as compared to parts located in the radiation shadow or ’leeside’. Therefore two integral coefficients are calculated: (I) (S) 1 C (C )= cτ(r) dS (4.30) F F ( ) Aref SI SS where SI (SS) denotes the illuminated (shadowed) part of the surface for given radiation attack angles, and cτ(r)=−2 (r) n(r)

For given temperatures TW,I, TW,S the coefﬁcients can be calculated (in Phase 2) by 4 4 C (α β )=C(I)(α β) TW,I + C(S)(α β) TW,S F , , TW,I, TW,S F , F , Tre f Tre f

(I) (S) (I) (S) with CF , CF (and CM , CM ) computed and stored in the database during Phase 1. In the ANGARA software system the spacecraft geometry is constructed from elementary geometrical elements. The surface of these elements is divided into small triangular panels. These panels represent the elementary surface elements, and the integrals are substituted by sums over all panels

N C = 1 c F ∑ τ,iSi (4.31) Aref i N C = 1 (r − r ) × c M ∑ i cm τ,iSi (4.32) Aref Lref i c The local aerodynamic or radiation stress coefﬁcients τ,i for panel number i are calculated as explained above. The spacecraft geometry is provided by the spacecraft model- r r r ing system as an array of panel corner points 1,i, 2,i, 3,i. The panel normal direction is taken as the direction of the vector s =(r − r ) × (r − r ) n 2 1 3 1 = |s | n = s |s | The panel area is given by S n /2 and the panel normal by n/ n . The local tangent direction is given by (with u = u(α, β) given earlier) t = n × (u × n)

4.4 Summary

To summarise, the ANGARA Monte-Carlo Test Particle method was used for radiation pressure and aerodynamic force coefﬁcient calculations, because of this method’s ability to represent the effects of multiple reﬂections at GOCE’s concave corners. The integral method was only used for validation of the Monte-Carlo results. The Monte-Carlo

42 force coefﬁcient calculation results were stored in matrices on disk, which are read in the NRTDM software. The NRTDM software reads these matrices, and interpolates for the attack and side-slip angles, and speed ratio, and corrects for the spacecraft wall temperature. The resulting aerodynamic and radiation pressure force coefﬁcient vectors are used in the algorithms described in Chapter 6.

43 44 Chapter 5 Reference frame transformations

Most of the calculations used in this project are performed in the satellite body-ﬁxed reference frame. But the input and output data make use several other reference frames, which are described in Section 5.1. The rotation matrices which are used for the transformations between these frames are presented in Section 5.2, while the conversion between various satellite attitude representations (quaternions, Euler angles, rotation matrix) are brieﬂy described in Section 5.3.

5.1 Reference frames

The six main reference frames which are used in the project’s algorithms are described below.

5.1.1 Local The local reference frame has its principal axes pointing in the East, North and up direction, respectively. This reference frame is used for studying wind vectors. The Horizontal Wind Model (HWM) output is provided in this frame, even though the up-component is always zero. Note that the axes of the local reference frame in this study are aligned with the principal axes of the Local North Oriented Frame (LNOF) as deﬁned in the GOCE standards document [Gruber et al., 2010]. However, in the GOCE LNOF, the order of the North and East axes has been switched, and the frame is therefore not right-handed. The local frame used in this study is a right-handed system, with the X-axis pointing East.

5.1.2 Earth-ﬁxed The Earth-ﬁxed frame has its origin in the Earth’s centre of mass, its Z-axis in the direction of the Earth’s rotation axis and its X-axis in the direction of the intersection of the equator

45 and Greenwich meridian. The reference system is therefore co-rotating with the Earth. The International Terrestrial Reference Frame 2005 (ITRF 2005) is a realization of this Earth-ﬁxed frame, based on various geodetic measurements. The ITRF 2005 is used in the GOCE orbit ephemeris calculations [Gruber et al., 2010], of which the results are used in this study.

5.1.3 Inertial

The inertial frame, or ICRF also has its origin in the Earth’s centre of mass. Its X-axis is closely aligned with the direction of the vernal equinox, while the Z-axis is closely aligned with the Earth’s rotation axis. Its realization is based on the observation of various celes- tial radio sources and space geodetic tracking of selected satellites. These observations are processed into values for precession, nutation, polar motion and TAI-UT1, which together with models provide the transformation between the Earth-fixed and inertial frames. In the GOCE standards document [Gruber et al., 2010], the inertial frame is dicussed in Section 4.1.2. In this project, the inertial frame is used in the computation of Keplerian elements, solar radiation pressure forces, and as an intermediate frame in the conversion of positions from the Earth-fixed frame to the spacecraft body-fixed frame, and of accelerations in the opposite direction.

5.1.4 Orbit-ﬁxed

For a perfectly circular orbit, the orbit-ﬁxed reference frame has its X-, Y- and Z-axes pointing in the inertial along-track, cross-track and radial directions, respectively. For eccentric orbits, the along-track and radial direction are not perpendicular (except at perigee and apogee). Therefore, one of these two directions is picked as the principal axis and the other is computed from the vector cross product of the chosen direction and the cross-track direction, in order to arrive at an orthogonal set of axes. For this study, the X-axis is chosen to be truly along-track (in the direction of the orbital velocity vector), and the Z-direction is perpendicular to both the along-track and cross-track directions. In the GOCE standards document [Gruber et al., 2010], this reference frame is called the Local Orbit Reference Frame (LORF).

5.1.5 Body-ﬁxed

The satellite body-fixed frame’s axes are attached to the spacecraft geometry (see Fig- ure 5.1. The body-fixed X-axis is in the length direction of the satellite, nominally pointing in the flight direction, which is the front of the satellite. Using this definition, the ion engine is at the back of the satellite, and provides its thrust mainly in the -X direction. The Y-axis is normal to the solar arrays on the satellite’s wings, nominally pointing in the direction of the Sun. The Z-axis is directed orthogonally to the smaller winglets at the back of the satellite. To complete the right-handed coordinate system, it nominally points towards the Earth. In the GOCE standards document [Gruber et al., 2010], this frame is known as the Gradiometer Reference Frame (GRF). Within this project, the calculations of aerodynamic

46 Along-track Radial Velocity

Sun SBF-X SBF-Y Pitch Roll

Cross-track

Ion engine thrust Yaw

SBF-Z Earth

Figure 5.1 Definition of the GOCE spacecraft body-fixed (SBF) axes and Euler angles. The orbit-fixed axes (along-track, cross-track and radial) are also shown for zero Euler angles. The alignment of the directions of these axes with the indicated Earth and Sun directions is approximate, and varies over time.

47 and solar radiation pressure accelerations on the satellite body are performed in this frame, which ultimately lead to the determination of density and wind.

5.1.6 Pseudo body-fixed The pseudo body-fixed frame is an intermediate frame, used for the calculation of Euler angles. This frame is aligned with the body-fixed frame, but the directions of the principle axes are interchanged so that these are close to the directions of the orbit-fixed frame, when the satellite is in a nominal attitude mode. In other words, the pseudo body-fixed frame is identical to the orbit-fixed frame when the Euler angles which describe the satellite attitude with respect to the orbit-fixed frame are zero.

5.2 Rotation matrices

Transformations between the different reference frames are achieved by making use of rotation matrices. The way these rotation matrices are calculated will be described here.

5.2.1 Local to Earth-ﬁxed

The rotation matrix Rle from the local to the Earth-ﬁxed frame is calculated using the local longitude φ and latitude λ (see Section 6.2.2) ⎛ ⎞ − sin(φ) − sin(λ) cos(φ) cos(λ) cos(φ) = ⎝ (φ) − (λ) (φ) (λ) (φ) ⎠ Rle cos sin sin cos sin (5.1) 0 cos(λ) sin(λ)

5.2.2 Inertial to earth-ﬁxed The computation of the inertial to Earth-ﬁxed rotation matrix is quite involved, including several intermediate steps for precession, nutation, Earth rotation and polar motion. The procedure is described in the IERS conventions [IERS, 2010], and implemented in the NRTDM software using the IAU’s SOFA subroutine library in Fortran [IAU, 2012].

5.2.3 Inertial to orbit-ﬁxed

The columns of the rotation matrix from the inertial frame to the orbit-fixed frame Rio are e e the inertial unit vectors in the along-track â, cross-track ˆc and (pseudo-)radial directions e ˆr. These unit vectors are computed from the orbit as follows: r˙ r × r˙ eˆ = ,ê = ,ê = eˆ × eˆ (5.2) a |r˙| c |r × r˙| r a c

5.2.4 Pseudo body-fixed to body-fixed The rotation matrix from the body-fixed frame to the pseudo orbit-fixed frame depends on the nominal orientation of the satellite with respect to the orbits along-track, cross- track and radial directions. For GOCE, this orientation is fixed, but orientations of the CHAMP and GRACE satellites was changed at various points in time, when mission op- erations decided to fly the satellites backwards with respect to their previous orientation.

48 Euler angles to rotation matrix http://www.j3d.org/matrix faq/matrfaq latest.html#Q36 Rotation matrix to Euler angles http://www.j3d.org/matrix faq/matrfaq latest.html#Q37 Quaternion to rotation mtrix http://www.j3d.org/matrix faq/matrfaq latest.html#Q54 Rotation matrix to quatenion http://www.j3d.org/matrix faq/matrfaq latest.html#Q55

Table 5.1 URLs of the Matrix and Quaternions FAQ, for the description of the various conversion algorithms between Euler angles, rotation matrices and quaternions

Since the body-ﬁxed X-axis for GOCE nominally points in the along-track direction, and the Z-axis nominally points opposite to the radial direction (see Figure 5.1), this rotation is achieved by simply ﬂipping the signs of the Y- and Z-directions: ⎛ ⎞ 100 = ⎝ − ⎠ Rbp 0 10 (5.3) 00−1

5.2.5 Inertial to body-ﬁxed q The rotation matrix Rib is obtained from the attitude quaternions , using the algorithm described in Section 5.3.

5.3 Quaternions, Euler angles and rotation matrices

The attitude data for GOCE is provided in the form of quaternions. The calculations relying on this attitude information are performed in NRTDM using rotation matrices. In addition, Euler angles are used to interpolate data gaps in the attitude information, and to plot the information for checking the data. The software therefore needs to convert between these two representations of the attitude information. The conversion algorithms between these three attitude representations were taken from the “Matrix and Quaternions FAQ”, maintained by Andreas Junghanns and hosted on the Java 3D community website www.j3d.org. These will not be repeated here. Table 5.1 provides the URLs to the relevant sections in the FAQ:

49 50 Chapter 6 NRTDM data importing and (pre)processing routines

The NRTDM software, which was originally developed under ESA contract for the processing of Two-Line Element orbit data [Doornbos, 2006] was later expanded for the processing of accelerometer data into thermosphere density and winds [Doornbos et al., 2009]. It has since undergone further improvements, in preparation for the processing of GOCE and Swarm measurements. The software is based on a centralized data storage, into which input products can be imported. These products can then be processed into derived products. These processing results are saved in the same storage system, and can therefore always be retrieved later for further analysis. By chaining and branching a number of these processing steps, all producing intermediate products, a set of end products can be created, in this case consisting of thermosphere density and wind data. Some of the intermediate products, such as geodetic coordinates and model data, can be used as auxiliary data when exporting the end products. This data export can be in the form of formatted ASCII data files, or in the form of NASA CDF files. The possibility of exporting NetCDF data files will be implemented in the near future. This Chapter will describe the importing and processing routines, which together encapsulate nearly all of the algorithms required for this project.

6.1 Data import

The following Sections describe the routines used to import external data into the NRTDM processing system.

6.1.1 SP3FileToOrbitFiles The Precise Science Orbits (PSO) for GOCE come in the SP3 ﬁle format. The Sp3FileToOrbitFiles routine reads this format and stores the orbit information in the NRTDM data storage

51 system.

Inputs • GO SST PSO

Outputs • Cartesian orbit positions in the Earth-ﬁxed frame (r)

• Cartesian orbit velocities in the Earth-ﬁxed frame (¨r)

Algorithm The routine encompasses a parser for the SP3 orbit ﬁle format, and outputs the orbit ephemeris data into the NRTDM data storage.

6.1.2 ImportASCIIFormatted All GOCE products, except for the GO SST PSO product come as formatted ASCII ﬁles, or can be converted to such a format using a simple parsing script. These data are imported into NRTDM using the ImportAsciiFormatted routine.

Inputs • GO EGG CCD

• GO EGG IAQ

• GOCE-Mass-Properties-COG-data.GOC

• Thruster activation data

Outputs • r Uncalibrated input accelerations (¨raw)

• Attitude quaternions describing the rotation between the inertial and spacecraft body-ﬁxed reference frame (q)

• Spacecraft mass (m)

• Thrust level of the ion thruster (T)

Algorithm This routine requires the a FORTRAN format specifier, to map the columns in the input file to the destination data fields. The routine can also resample the data by using a simple linear interpolation of the input data.

52 6.1.3 PolynomialCalibrationParameters The common-mode accelerometer calibration biases can usually be well approximated by polynomials as a function of time. The PolynomicalCalibrationParameters input routine evaluates such polynomials at the speciﬁed time intervals, in order to get the values into the NRTDM data storage for further computations.

Inputs • Start dates of polynomial ﬁt intervals • Polynomial coefﬁcients for each of the biases and scale factors

Outputs • p Calibration scale factors ( s) • p Calibration biases ( b)

Algorithm The routine does nothing more than evaluate the speciﬁed polynomials at the requested times and write the results into the NRTDM data storage.

6.2 Basic conversions

After import, described in the previous Section, the processing of data starts with several basic conversions. Figure 6.1 shows the flow chart, which basically consists of four independent chains: • Conversion of Cartesian orbit coordinates to geodetic coordinates; • Computation of relative velocity components in the body-fixed frame due to the orbital velocity and atmospheric coration velocity, after filling gaps in the attitude quaternions through the conversion to Euler angles and back; • Initial calibration of the accelerometer data; • Conversion of thruster activation data from forces to accelerations. The algorithms taking place within the processing subroutines, indicated with blue boxes in the flow chart, will be discussed in turn below.

6.2.1 CalibrateAccelerometer The accelerometer data is calibrated by applying biases and scale factors.

Inputs • r Uncalibrated input accelerations (¨raw) • p ) Calibration scale factors ( s • p Calibration biases ( b)

53 SST_PSO EGG_IAQ

SP3FileToOrbitFiles ImportASCIIFormatted

Orbit_PSO Quat_EGG_IAQ

OrbitToGeo WindCorotationSBF

Orbit_Geo Orbit_SBF_Velocity

GOCE-Mass- Calibration using AUX_NOM EGG_CCD Properties-COG-data GPS

XML parser Polynomial ﬁt script in product metadata

ImportASCIIFormatted ImportASCIIFormatted ImportASCIIFormatted PolynomialCalibrationParameters

SpacecraftMass IonThruster Accel_EGG_CCD Accel_CalibrationParameters

GOCEThrusterAccel CalibrateAccelerometer

Accel_IonThruster Accel_Calibrated

Figure 6.1 Flow chart of the data import and preprocessing in the GO Basic product category.

54 Outputs • r Calibrated output accelerations (¨cal)

Algorithm The calibration biases and scale factors are applied as follows: ⎛ ⎞ ps,xr¨raw,x r = ⎝ ⎠ + p ¨cal ps,yr¨raw,y b (6.1) ps,zr¨raw,z

6.2.2 OrbitToGeo The OrbitToGeo algorithm computes geodetic and geomagnetic coordinates as well as the argument of latitude from the Cartesian Earth-ﬁxed orbit ephemeris information. The output is used as input in density and wind model evaluation, and can be useful for generating plots as well.

Inputs The following NRTDM internal input data products are required: • Cartesian orbit positions in the Earth-ﬁxed frame (r)

Outputs The output product contains the following ﬁelds: • Altitude (h)

• Geodetic longitude (φ)

• Geodetic latitude (λ)

• Local solar time (lst)

• Argument of latitude (u) • φ Geomagnetic longitude ( m) • λ Geomagnetic latitude ( m) • Geomagnetic local solar time (lstm)

Algorithm This routine uses the algorithm found in Montenbruck and Gill [2000, Section 5.5] for the conversion from Cartesian coordinates to geodetic coordinates. The local solar time is φ computed from the longitude and time of day in seconds tsec as follows: t φ lst = sec + (6.2) 3600 15 The argument of latitude is calculated in two steps:

55 1. The Earth-fixed orbit positions and velocities are converted to the true of date inertial reference frame Montenbruck and Gill [2000, Chapter 5]; 2. The argument of latitude u is found as part of the conversion of true of date inertial positions velocities to Keplerian elementsMontenbruck and Gill [2000, Section 2.2.4]; The equivalent geomagnetic coordinates are computed by first making use of the GE- OMAG 08 routine from the GEOPACK 08 library, in order to convert the Cartesian Earth- fixed coordinates to geomagnetic coordinates.The GEOPACK 08 library can be found at http://geo.phys.spbu.ru/∼tsyganenko/Geopack-2008.html.

6.2.3 WindCorotationSBF The WindCorotationSBF takes the Earth-ﬁxed Cartesian orbit ephemeris, and computes the relative velocity of the atmosphere with respect to the spacecraft, due to the orbital velocity and due to the atmospheric corotation velocity.

Inputs The following NRTDM internal input data products are required: • Cartesian orbit positions in the Earth-fixed frame (r) • Cartesian orbit velocities in the Earth-fixed frame (˙r) • Attitude quaternions describing the rotation between the inertial and spacecraft body-fixed reference frame (q)

Outputs The output product contains the following data: • v Relative velocity due to orbital velocity ( o) • v Relative velocity due to atmosphere corotation ( c)

Algorithm The two relative velocity components are computed using the following equations: v = − r o Rib ˙ (6.3)

v = (ω × r) c Rib ⊕ (6.4) ω where ⊕ is the Earth’s angular velocity vector and Rib is the rotation matrix from the inertial to the satellite body-ﬁxed frame, computed as a function of the quaternion q

6.2.4 QuaternionToEulerAnglesConvert Euler angles, which are derived from quaternions with the help of orbit information, can be used for inspecting the satellite attitude behaviour. They are not essential to the further data processing, and were therefore not included in Figure 6.1

56 Inputs

• Attitude quaternions describing the rotation between the inertial and spacecraft body-ﬁxed reference frame (q)

• Cartesian orbit positions in the Earth-ﬁxed frame (r)

• Cartesian orbit velocities in the Earth-ﬁxed frame (˙r)

Outputs

• Roll angle (Φ)

• Pitch angle (Θ)

• Yaw angle (Ψ)

Algorithm

The Euler angles are computed from the rotation matrix Rbp between the body-ﬁxed frame and the pseudo body-ﬁxed frame, which can be computed by multiplying three intermediate rotation matrices.

= Rbp RbpRibRoi (6.5)

These intermediate matrices are described in Section 5.2. From this rotation matrix, the Euler angles can be computed as described in Section 5.3.

6.2.5 GOCEThrusterAccel The ion thruster activation data is provided in the form of a scalar value, representing the thrust force. Here it is converted to an acceleration vector, so that it can subsequently be removed from the common-mode accelerometer measurements.

Inputs

The following input data products are required

• Ion thruster activation level (T)

• Satellite mass (m)

The following metadata keys are required

• α Z cross-coupling angle ( T)

Outputs • a Ion thruster acceleration ( T)

57 Orbit_Geo Quat_EGG_IAQ

GeoToDens GeoToWind

DensityModel_NRLMSISE-00 WindModel_Local_HWM07

WindLocalToSBF

WindModel_SBF_HWM07

Figure 6.2 Flow chart of the GO Models product category, containing evaluated thermosphere model data.

Algorithm

The thrust acceleration is computed from the reported thrust force. It is turned into a vector by assuming the thrust force is in the X-Z plane, predominantly in the +X-direction, α with a small deﬂection angle T providing a component in the Z-direction.

⎛ ⎞ α cos T r = T ⎝ ⎠ ¨T 0 (6.6) m α sin T

6.3 Thermosphere model evaluation

The evaluation of two thermosphere models, NRLMSISE-00 and HWM07, is part of the nominal processing. The ﬂow-chart for this part of the processing is provided in Fig- ure 6.2. The calculation of density model values, based on time and location, is straightforward. Since the wind is a vector quantity, quaternions are required after the model evaluation, to convert it to the body-ﬁxed frame.

58 6.3.1 GeoToDens Inputs The density model requires the location in geodetic coordinates, which is provided as an input product:

• Altitude (h)

• Geodetic longitude (φ)

• Geodetic latitude (λ)

• Local solar time (lst)

In addition, the solar and geomagnetic activity proxies are retrieved as a function of time.

Outputs • Total neutral density (ρ) ρ • i + Mass concentrations of the constituents ( ρ , where i = H, He, O, O ,N,O2,N2, Ar)

• Temperature (T)

Algorithm The GeoToDens subroutine just gathers the necessary input data, converts it to the right data types, calls the model subroutine, and converts the outputs to the right data types.

6.3.2 GeoToWind The GeoToWind routine is very similar to the GeoToDens routine. It calls the HWM07 model, and returns the wind vector in the local coordinate frame.

Inputs The wind model requires the location in geodetic coordinates, which is provided as an input product:

• Altitude (h)

• Geodetic longitude (φ)

• Geodetic latitude (λ)

• Local solar time (lst)

In addition, the solar and geomagnetic activity proxies are retrieved as a function of time.

Outputs • v Wind vector in the local (East, North, up) frame ( w,local).

59 Algorithm The GeoToWind subroutine just gathers the necessary input data, converts it to the right data types, calls the model subroutine, and converts the outputs to the right data types.

6.3.3 WindLocalToSBF This routine converts a vector in the local reference frame, with axes in the East, North and up directions, to the spacecraft body-ﬁxed reference frame. This conversion is used on the Horizontal Wind Model output, for which the up-component is by deﬁnition always zero.

Inputs • v Wind vector in local (East, North, up) reference frame ( w,local) • Geodetic longitude (φ)

• Geodetic latitude (λ)

• Attitude quaternions describing the rotation between the inertial and spacecraft body-ﬁxed reference frame (q)

Outputs • v Wind vector in the spacecraft body-ﬁxed reference frame ( w,sbf)

Algorithm The rotation of the vector from the local (index l) to the body-ﬁxed (index b) frame is done using the Earth-ﬁxed and Inertial (indices e and i) intermediate reference frames.

v = v = v w,sb f Rlb w,local RibReiRle w,local (6.7) Refer to Section 5.2 for details on the computation of the intermediate rotation matrices.

6.3.4 WindSBFtoLocal Inputs • v Wind vector in the spacecraft body-ﬁxed reference frame ( w,sbf) • Geodetic longitude (φ)

• Geodetic latitude (λ)

• Attitude quaternions describing the rotation between the inertial and spacecraft body-ﬁxed reference frame (q)

Outputs • v Wind vector in local (East, North, up) reference frame ( w,local)

60 Algorithm The algorithm is just the opposite to that of the WindLocalToSBF routine. The conversion between reference frames is once again done through the intermediate inertial and Earth- ﬁxed frames.

v = v = v w,local Rbl w,sb f RelRieRbi w,sbf (6.8) Refer to Section 5.2 for details on the computation of the intermediate rotation matrices.

6.3.5 WindVectorProject Only the crosswind component can be determined from the accelerometer data. In order to be able to directly compare this data with wind data from other sources, such as the HWM07 model, the HWM07 wind data must be projected on the crosswind.

Inputs • v Model wind ( w,hwm) • v Crosswind measurement ( w,cr)

Outputs • v Projected crosswind component of the model wind ( w,hwm,cr)

Algorithm First, the unit vector in the crosswind direction is determined: v v = w,cr ˆ w,cr |v | (6.9) w,cr

Then the projection of the model wind on this unit vector is determined using the dot product: v =(v · v )v w,hwm,cr w,hwm ˆ w,cr ˆ w,cr (6.10)

6.4 Force model evaluation

6.4.1 RadiationPressureProduct

Inputs • Cartesian orbit positions in the Earth-ﬁxed frame (r)

• Attitude quaternions describing the rotation between the inertial and spacecraft body-ﬁxed reference frame (q)

• Spacecraft mass (m)

61 Orbit_PSO Quat_EGG_IAQ SpacecraftMass

RadiationPressureProduct

Accel_Calibrated RadiationPressure Accel_IonThruster

AccelerometerSubtract

Accel_Aero_Calibrated

Figure 6.3 Flow chart for part of the GO Panels product category, containing non- gravitational force model results and removal of unwanted signals from the accelerometer data for density and wind determination.

62 Outputs • r Acceleration due to solar radiation pressure (¨srp) • r Acceleration due to Earth albedo radiation pressure (¨alb) • r Acceleration due to Earth IR radation pressure (¨IR)

Algorithm overview An overview of radiation pressure calculations is provided in Chapter 4

Varying Sun-Earth distance algorithm For a realistic solar radiation pressure calculation, the varying distance of the spacecraft from the Sun has to be taken into account. For sunlight at a distance of 1 astronomical unit (1AU = 149, 597, 870, 660 m) from the Sun, the radiation pressure has a value of = −6 2 P1AU 4.56 10 N/m . Taking the varying Sun-satellite distance rs into account and applying the inverse square law as well as the shadow function from Equation (6.17), the solar radiation pressure can be expressed as 2 = 1AU P fsP1AU . (6.11) rs

Eclipse algorithm For a satellite in a low-Earth orbit, the Sun can often be eclipsed by the Earth. Less often, the Moon can cause eclipses as well. In order to model the inﬂuence of eclipses on the radiation pressure, a shadow function fs is introduced. The shadow function is deﬁned to have a value of one when the satellite is in full sunlight, and zero when the satellite is ◦ fully eclipsed. Because the Sun forms a solid disc of about 0.5 , the Earth and Moon cast a semi shadow region, or penumbra around the core shadow, or umbra. The discussion below will explain the calculation procedure for eclipses by the Earth. The calculation for eclipses by the Moon can of course be made in much the same way by substituting lunar coordinates for the Earth-centred coordinates, and by disregarding the atmospheric effects on the eclipse. Different algorithms exist in order to determine whether a satellite is in eclipse [Mon- tenbruck and Gill, 2000, Vallado, 2001, Adhya et al., 2004]. An especially elaborate treatment of the problem can be found in the series of papers by Vokrouhlicky´ et al. [1993, 1994a, 1996]. The method outlined below is provided in the ANGARA level 2 software [Fritsche et al., 1998]. It can be used to describe atmospheric effects on the semi-shadow boundaries. The basis of this method is a conical shadow model, where the Sun’s disc is projected on an Earth radius vector which is perpendicular to the Sun-satellite line (see Figure 6.4). The subscript symbols ⊕ and are used to indicate positions with respect to the centres of the Earth and Sun, respectively, while the subscript s can be used to denote r r the satellite position if this is not already obvious. Let and s then denote the position vectors of the satellite with respect to the centres of the Earth and Sun. The vector r = r · r r ps (ˆs ⊕s) ˆs (6.12)

63 a) b)

Sun

Earth Sun Earth

penumbra

umbra Earth

Figure 6.4 The conical eclipse model, showing (a) a schematic view from the satellite, (b,c) the geometric relationship between the Earth, Sun, the satellite at point s, and the projection of the Sun-satellite rays at the Earth closest point at point p.

64 a)Sun b)

Earth Sun Earth

Figure 6.5 (a) Distortion of the shape of the solar disc due to atmospheric refraction as seen from the satellite (schematic). (b) Eclipse transition geometry under the inﬂuence of atmospheric refraction.

then deﬁnes the point p, which is the closest point to the Earth’s centre on the Sun-satellite vector. This vector points from p to the satellite s in Figure 6.4. The vector from the Earth’s centre to p is r = r − r ⊕p ⊕s ps (6.13) The height above the Earth’s surface of a line connecting the center of the Sun with the satellite is then equal to = ||r || − hg ⊕p R⊕ (6.14) and the apparent radius of the solar disc projected on a plane through this point, perpendicular to the satellite-Sun vector, as indicated in Figure 6.4, is |r | = ps Rp |r | R (6.15) s In these equations, R⊕ and R are the radii of the Earth and Sun, respectively. In the currently implemented version of the algorithm, these radii are considered constants. An extension which takes the oblateness of the Earth into account should improve the accuracy of the calculations [Vokrouhlicky´ et al., 1996, Adhya et al., 2004]. The geometric heights above the Earth’s surface of the centre, top and bottom ray = = + = − of light from the solar disc are hc hg, ht hg Rp and hb hg Rp, respectively. A negative value indicates that the beam is blocked by the Earth (see Figure 6.4). The apparent height of these rays, as seen from the satellite will differ from these values, due to refraction in the atmosphere. As shown in Figure 6.5, refraction will lead to a narrowing of the full eclipse cone. An elaborate discussion can be found in Vokrouhlicky´ et al. [1994b]. An auxiliary parameter η can be introduced such that

η = hc − , (6.16) hc hb where

η < −1 if the satellite is in full eclipse −1 ≤ η ≤ 1 if the satellite is in partial eclipse η > 1 if the satellite is in full sunlight

65 Figure 6.6 Example of a monthly ANGARA Earth albedo map (left) and an Earth infrared radiation map, based on ERBE data. The shade of grey is proportional to the percentage of sunlight that is reﬂected, or the infrared radiation that is emitted by the Earth surface elements, respectively.

Two effects can be recognised that determine the value of the shadow function: geometric shadowing of part of the solar disc by the solid Earth and absorption of sunlight by the Earth’s atmosphere. The shadow function can be expressed as the product of these two contributions. = fs fg fa (6.17)

The geometric shadowing factor fg indicates the fractional area of the solar disc that is blocked by the Earth. Its calculation can be found in Carrou [1995] or Montenbruck and Gill [2000], resulting in the following formula: η = − 1 (η) + − η2 fg 1 π arccos π 1 (6.18) The amount of sunlight that is not absorbed in the Earth’s atmosphere is represented by the factor fa. Absorption of sunlight in the atmosphere will lead to an extension of the semi-shadow region [Vokrouhlicky´ et al., 1994b]. The precise modelling of the effects of absorption, refraction and the elliptical shape of the Earth on the eclipse have not yet been included in the current version of the processing software.

6.4.2 ANGARA Earth radiation pressure environment model Because of the proximity of the satellite to the surface of the Earth, we can not treat the Earth as a single light source. Therefore, the Earth radiation pressure accelerations are calculated by summation of the contributions of a number of Earth elements. The ANGARA software provides detailed Earth maps of Earth Radiation Budget Experiment (ERBE) satellite data (see Figure 6.6). These bitmaps have a resolution of ◦ ◦ 144 × 72 pixels, corresponding to surface elements of 2.5 × 2.5 . The ERBE data, span- ning several years in the late 1980’s, are available in the form of monthly averaged maps containing albedo coefﬁcients and infrared intensity data. The Earth albedo and infrared radiation maps are stored in bitmap ﬁles in equidistant ◦ cylindrical projection, with the top left corner of the map corresponding to +90 latitude ◦ and -180 longitude. Figure 6.7 shows various parameters involved in the following calculations.

Figure 6.7 Deﬁnition of geometrical parameters in the computation of Earth radiation pressure.

If the map has a resolution of L × M pixels and ◦ 360 Δλ = (6.19) L ◦ 180 Δφ = (6.20) M then each pixel corresponds to an Earth surface element measuring Δλ × Δφ degrees. If a certain pixel is in column m and row l (with 1 ≤ m ≤ M and 1 ≤ l ≤ L), the φ λ longitude k and latitude k of the center of the corresponding Earth surface element with index k = m(l − 1)/L can be found from φ = ◦ − ( − )Δφ k 180 l 0.5 (6.21) λ = − ◦ +( − )Δλ k 90 m 0.5 (6.22) These coordinates can easily be transformed into Earth-ﬁxed Cartesian coordinates, and r subsequently to spacecraft body-ﬁxed coordinates, which then form the vector k. The area of the Earth surface element is a function of the map resolution and the element’s latitude. 4πR2 π/2 (m − 0.5) π A = E sin sin (6.23) k L M M

67 In order to find the albedo radiation pressure at the satellite position Palbk from a surface element k, we first need to find the incoming radiation pressure from the Sun on this element. This value can be found from the radiation pressure at a mean distance of 1 AU, corrected for the actual distance between the Sun and the surface element. 2 = 1AU P,k P1AU |r − r | (6.24) k The percentage of this radiation that actually arrives at the satellite depends on the albedo coefficient Calb and on the elevation angles of the Sun E and the satellite Esat. These two angles can be found from the following unit vectors, pointing from the surface element to the Sun, the satellite and the surface normal, respectively: r − r r − r r u = k u = k u = k k |r − r |, ks |r − r |, ⊥k |r | (6.25) k k k The solar and satellite elevation angles for the surface element can then be found using = u · u sin E,k k ⊥k (6.26) = u · u sin Esat,k ks ⊥k (6.27) The albedo radiation pressure for the surface element is zero if either the satellite or the Sun have a negative elevation angle. Otherwise the sunlight will be reflected following the Lambert law for a diffuse body, and decrease with distance according to the inverse square law, resulting in the following expression: < < 0 if sin Esat,k 0 or sin E,k 0 Palb = (6.28) k alb sin Esat,k 1 > > C AkP,k sin E,k π |r −r|2 if sin Esat,k 0 and sin E,k 0 k k

IR The Earth infrared radiation pressure for each Earth element k, denoted by P⊕,k can be read directly from the map. The resulting radiation pressure at the satellite position then becomes < 0 if sin Esat,k 0 PIR = (6.29) k IR sin Esat,k 1 > AkP⊕ π |r −r|2 if sin Esat,k 0 ,k k Since for each of the Earth’s surface elements, the vector to the satellite is different, a summation should take place to determine the accelerations. Note also that the op- tical properties of the satellite for visible light (albedo) and infrared light are different, ( ) requiring separate calculations of CF A .

L×M r = Aref Calb| alb ¨alb ∑ F kPk (6.30) m k=1 L×M r = Aref CIR| IR ¨IR ∑ F kPk (6.31) m k=1

6.4.3 AccelerometerSubtract This routine removes acceleration components from the measured and calibrated common- mode accelerations.

68 Inputs • r Calibrated common-mode accelerations (¨cal) • r Ion thruster accelerations (¨ion) • r Solar radiation pressure accelerations (¨srp) • r Earth albedo radiation pressure accelerations (¨alb) • r Earth infrared radiation pressure accelerations (¨cal)

Outputs • r Aerodynamic accelerations (¨aer)

Algorithm The algorithm consists of a simple subtraction of accelerations: r = r − r − r − r − r äer ¨cal ïon ¨srp älb ïr (6.32)

6.5 Density and wind processing

Figure 6.8 shows the density and wind processing ﬂow chart. Two algorithms are used for this: 1. A direct algorithm, providing density from the SBF-X-component only;

2. An iterative algorith, providing density and crosswind from the SBF-X and SBF-Y- directions combined. The direct approach is used for an initial density determination. This density is used for calibration of the Y-axis accelerations using models. Once the Y-axis acceleration is properly calibrated, the iterative algorithm can be used. These algorithms will be described in detail below.

6.5.1 DensityWindFromAccelerometerDirect Inputs • Calibrated common-mode accelerations, aerodynamic component only (r¨aer) • Relative velocity in the SBF frame due to the orbit (vo) • Relative velocity in the SBF frame due to orbit corotation (vc) • Modelled wind (vw) ρ i • Modelled mass concentrations of the atmospheric constituents ( ρ )

• Modelled thermosphere temperature (T)

• Spacecraft mass (m)

69 DensityModel_NRLMSISE-00 WindModel_SBF_HWM07

Accel_Aero_Calibrated

SpacecraftMass Orbit_SBF_Velocity

DensityWindFromAccelerometerDirect

Density_Direct Wind_SBF_Direct Aero_YZModelProjectedFromXDens

CalibrateWithModel

Accel_Aero_Calibrated_FromModel Accel_CalibrationParameters_FromModel

DensityWindFromAccelerometerIterative

Density_Iterative Wind_SBF_Iterative

WindSBFtoLocal

Wind_Local_Iterative Wind_Local_HWM07 Wind_SBF_HWM07

WindVectorProject WindVectorProject

Wind_Local_HWM07_Projected Wind_SBF_HWM07_Projected

Figure 6.8 Flow chart for part of the GO Panels and GO Results product categories, containing density and wind results.

70 Figure 6.9 Relative velocity, modelled and observed accelerations in the CHAMP spacecraft body-ﬁxed (SBF) XY plane. CHAMP is viewed from the top.

Outputs • Density (ρ) • Wind derived from the SBF-Y acceleration component (vw,y )

• Aerodynamic acceleration vector derived from the aerodynamic model, using the density derived from the acceleration X-component (r¨aer,proj)

Accelerations and velocities Figure 6.9 provides a simple schematic view of the three vectors of importance for a density and wind retrieval algorithm: The observed and modelled aerodynamic acceler- a a ations obs and mod are shown originating in the centre of mass of the satellite. In ad- v dition, the relative velocity r of the atmosphere with respect to the spacecraft is shown. This quantity is partly observed and partly modelled, as will be explained below. Note that due to the asymmetrical shape of the satellite with respect to the ﬂow, the acceleration is in general not exactly aligned with the relative velocity, as indicated by the dashed guide lines. The modelled and observed aerodynamic acceleration vectors initially do not match in magnitude and direction. This is mainly due to the approximate character of the modelled density and wind speed. It is the purpose of the new algorithm to ﬁnd those density and wind values, which, when replacing the original values, make these accelerations match. Before more detailed descriptions of the previous and new algorithms are provided, the relationship between the parameters in Figure 6.9 and the way in which they can be obtained from satellite observation data sets and models will be described. The description in the following Sections will refer to the instruments and data products of the current generation of accelerometer missions (CHAMP and GRACE) and to the external models (atmospheric models, force models, etc.) that are currently available. The new algorithm is not limited by the use of these data sets and models, however. It can be applied just as well to equivalent data from historical or future accelerometer missions and using future improvements to external models.

71 Relative velocity The relative velocity of the atmosphere with respect to the spacecraft is equal to the sum of contributions from the inertial velocity of the spacecraft in its orbit, the velocity caused by the corotating atmosphere, and the velocity of winds, with respect to an Earth-fixed atmosphere. For the accelerometer satellites, these can be expressed in satellite body- fixed (SBF) coordinates, as v = − v + (ω × r)+ v r Rib Rib ⊕ RibReiRle w , (6.33) in which the rotation matrix Rib from the inertial to the satellite body-fixed frame is obtained from star camera observations; the inertial satellite position and velocity, r and v, are obtained by precise orbit determination using tracking observations from the satel- ω v lite’s GPS receiver; ⊕ is the Earth’s angular velocity vector. The wind velocity w is usually specified in local coordinates (East, North, Vertical). The transformation to satellite body-fixed coordinates requires the additional rotation matrices Rle from local to the Earth-fixed Cartesian coordinate system, based on the satellite’s latitude and longitude [Montenbruck and Gill, 2000], and the rotation matrix Rei from Earth-fixed to inertial v coordinates, based on Earth orientation parameters.The first two contributors to r, the orbit and corotation velocities, are known at a much higher accuracy than the wind ve- v locity. If model values for w are required, these can be obtained from a wind model, such as HWM07[Drob et al., 2008]. In the description of the density and wind retrieval algorithms, we will use the nota- v v tion r,0 or r,i=0 to indicate an initial guess of the relative velocity, by either neglecting v winds or using a wind model. The notation r,i will designate a relative velocity which already includes an accelerometer-derived wind component.

Observed aerodynamic acceleration a The observed aerodynamic acceleration obs is obtained from the raw accelerometer data after calibration and removal of non-aerodynamic acceleration signals. Details on the accelerometer instruments of CHAMP and GRACE, their performance and processing of raw data can be found in Touboul et al. [2004]. The data is delivered to science users in the form of Level 2 products for CHAMPcitechampformat and Level 1B products for GRACE [Case et al., 2004]. Calibration is performed by adding a bias vector: a = a + a cal raw bias (6.34) The GOCE accelerations have already been corrected using scale factors, which were determined using special manoeuvres. This makes the calibration easier than for CHAMP and GRACE, for example. The bias determination was described in more detail in Chap- ter 3. Various non-aerodynamic signals should be removed from the accelerometer data, including accelerations due to activity of cold gas thrusters for attitude control. If a set of two opposing thrusters is not perfectly balanced, as is often the case, they introduce a residual signal in the linear acceleration, besides the intended angular acceleration. Data around the activation times of these thrusters should therefore be removed. A less obvious example of accelerations that should be removed from the data are those due to mechanical forces caused by electrical current changes on the satellite [Flury et al., 2008].

72 Figure 6.10 Schematic representation of the determination of density from the projection of acceleration on the XSBF axis.

a Finally, modelled accelerations due to radiation pressure from the Sun srp, Earth a a albedo alb and Earth infrared radiation IR are computed and removed from the cali- a brated and edited accelerometer data cal, to arrive at the observed aerodynamic acceler- a ation vector obs. a = a − a − a − a obs cal srp alb IR (6.35)

Modelled aerodynamic acceleration The modelled aerodynamic acceleration vector is expressed as:

A 1 a = C ref ρv2 (6.36) mod a m 2 r The mass m can be obtained by subtracting from the satellite launch mass the amount of propellant used, which is logged in the satellite’s housekeeping data, and made available as an auxiliary data product. Note that the value of Aref is an arbitrary constant, which C should be the same as the one that was used in the calculation of a (see Chapter 4).

Direct algorithm Previously published algorithms made use of assumptions about the orientation of the accelerometer in space. For GOCE, the accelerometer instruments are carefully mounted with their orientation so that their three axes can be considered perfectly aligned with the spacecraft body-ﬁxed (SBF) axes (see Section 5.1.5). The spacecraft are under active attitude control, which keeps these axes within a few degrees of the orbit-ﬁxed along- track, cross-track and radial directions (see Figure 5.1). The relative orientation of these axes can be expressed in roll, pitch and yaw Euler angles. Because these Euler angles are relatively small, the inertial orbital velocity of the satellite is kept closely aligned with the XSBF axis. Accelerations in the velocity direction are the most effective in changing the orbital energy, and therefore have a much larger effect on the orbit than accelerations of similar magnitude in perpendicular directions. This means that the XSBF axis of the accelerometer can be more accurately calibrated using positioning data from the GPS instrument [Van Helleputte et al., 2009] than the YSBF and

73 Figure 6.11 Schematic representation of the determination of wind from the accelerometer YSBF axis.

ZSBF axes, even without taking into account the larger measured signal. This consideration leads to an approach for density determination [Bruinsma and Biancale, 2003, Bruinsma et al., 2004, Sutton et al., 2007] where only the projection of the aerodynamic acceleration on the XSBF axis is used, as shown schematically in Figure ??. The density can then be solved directly from the X-component of the vector Equation (6.36):

ρ = 2m aobs,x 2 (6.37) Arefvr,0 Ca,x

Information from the acceleration component in the YSBF direction, closely aligned with the cross-track direction, can be used to derive data on the wind speed in that di- C rection. Sutton et al. [2007] describe two approaches. In the first approach, a in Equa- tion (6.36) is expanded using the analytical equations for a panel model.The resulting expansion is quadratic with respect to vw,y , which can then be solved, resulting in an ex- ρ pression depending on aobs,y and . Sutton names this approach the single axis method, even though information from both the X- and Y-axes is required, if ρ is to be substituted from Equation (6.37). This method is not compatible with the ANGARA force coefficient calculation adopted for this study. In addition, it is less accurate than other models, so it will not be further discussed here. The second approach is named the dual-axis method by Sutton et al. [2007], and can be found in an earlier paper by Liu et al. [2006] as well. The method requires that the lift and sideways forces are negligible, or are modelled and removed from the accelera- a tion beforehand, so that only the observed acceleration due to drag obs,D remains. The authors do not specify exactly how the lift and sideways forces should be modelled, but we have adopted the following approach: first, a new modelled aerodynamic acceleration is computed according to Equation (6.36), now with the density from Equation (6.37) v a and with the a-priori relative velocity r,0 as inputs. This acceleration vector mod can then be decomposed into a drag component, by projection on the relative velocity direction, and a perpendicular lift plus sideways force component, by subtraction of that drag component from the original modelled acceleration. In equations:

a =(a · v )v a = a − a mod,D mod ˆ r,0 ˆ r,0 , mod,L mod mod,D (6.38)

74 a The modelled lift plus sideways aerodynamic force mod,L is then subtracted from the observed aerodynamic acceleration, to arrive at the observed drag. a = a − a obs,D obs mod,L (6.39) The velocity and drag acceleration are by deﬁnition in the same direction, so that the wind can be determined from a simple geometrical consideration (see Figure 6.11). ρ C Expressed in the form of an equation, and D disappear when the Y-component of Equation (6.36) is divided by the X-component, and vw,y is solved for after substitution of Equation (6.33), resulting in: = aobs,D,y − vw,y vr,0,x vr,0,y (6.40) aobs,D,x

A similar wind determination could in principle be performed for the ZSBF axis. These wind computations are not used, however, because more accurate results can be obtained using the iterative algorithm, described below.

Discussion of the direct algorithms The schematic representations in Figures ?? and 6.11 clearly show that when the angle between the relative velocity and the XSBF axis gets larger, the errors in the density and wind increases. For the extreme case where this angle approaches 90 degrees, density values will approach zero, while the wind speed will go to infinity. In these figures, the angles are exaggerated for clarity, compared to the angles of GOCE under nominal attitude control. These attitude angles only determine the alignment of the body-fixed frame with the inertial velocity vector. The contributions to the relative velocity vector by the co-rotation of the atmosphere and thermosphere winds can be equally important. The atmospheric co-rotation velocity over the equator depends on the altitude, ranging from 483–502 m/s at 250–500 km. This increases the maximum angle between the relative velocity and the XSBF axis by 3.6–3.8 degrees. The wind speed, which under most conditions is within the range of about 0–200 m/s, can reach peak velocities in the polar regions of up to 500– 1000 m/s [Luhr¨ et al., 2007, Forster¨ et al., 2008], causing the incidence angle to reach peak values of 8–10 degrees. In principle, the accuracy of the derived density and wind speed should be independent of these angles, but when using the direct approach this is not the case. C Another limiting factor of the direct algorithm results from the dependence of a on v v r. The methods use an initial value r,0 , composed of the orbit and co-rotation velocity, and either neglect or model the in-track wind velocity. After the derivation of the cross- track wind vw,y however, there is a better estimate of the relative velocity: v = v + v r,i=1 r,0 w,y (6.41) where the index i is an iteration counter. This new relative velocity leads to a new value C of a (according to the equations in the previous Chapter) and therefore to a new value ρ C of . The change in a also leads to a change in the lift and sideways components of the a a aerodynamic acceleration, which are to be removed from obs to arrive at obs,D, yielding a new value for vw,y . This chain of dependencies indicates that an iterative algorithm could be required to determine the density and wind speed with the highest possible accuracy.

75 6.5.2 CalibrateWithModel

Inputs • Aerodynamic acceleration, Y- and Z-components not accurately calibrated (r¨aer,in)

• Aerodynamic acceleration vector derived from the aerodynamic model, using the density derived from the acceleration X-component (r¨aer,proj)

Outputs • Calibration biases (pb)

• ¨ Calibrated aerodynamic acceleration (raer,cal)

Algorithm

The calibration biases are determined by minimizing, in a least-squares sense, the differences between the two input accelerations, using data points from a sufﬁciently long time interval (e.g. a day). The Levenberg-Marquardt algorithm from the Numerical Recipes [Press et al., 1996] is used for the least-squares minimisation. The output calibrated aerodynamic accelerations are obtained by applying the new- found biases to the input non-calibrated aerodynamic accelerations.

6.5.3 DensityWindFromAccelerometerIterative

Inputs • Calibrated common-mode accelerations, aerodynamic component only (r¨aer)

• Relative velocity in the SBF frame due to the orbit (vo)

• Relative velocity in the SBF frame due to orbit corotation (vc)

• Modelled wind (vw)

ρ i • Modelled mass concentrations of the atmospheric constituents ( ρ )

• Modelled thermosphere temperature (T)

• Spacecraft mass (m)

Outputs

• Density (ρ)

• Crosswind (vw,cr)

Figure 6.12 Schematic overview of the iterative wind and density derivation algorithm for accelerometer satellites.

77 Algorithm This Section presents an iterative algorithm, which avoids the restrictions and sources of error discussed in the previous Section. Figure 6.12 illustrates schematically the principle of the algorithm in two steps. The goal of the algorithm is to make the modelled aerody- a namic acceleration mod match the direction (top panel) and subsequently the magnitude a (bottom panel) of the aerodynamic acceleration observed by the accelerometer obs. This v is achieved by first modifying the direction of the relative velocity vector r, without modifying its magnitude, until the modelled acceleration direction matches that of the observed acceleration. Subsequently the density ρ is modified, so that the magnitude of the accelerations matches. The adjustment to the orientation is made by a rotation of the relative velocity about e the local vertical direction, indicated by the unit vector ûp. The acceleration components projected on this direction will be set to zero. To simplify the notation, a prime is added to indicate this modification of the acceleration vectors, which is applied repeatedly:

a = a − (a · e ) e ˆup ˆup (6.42)

We will use the sum of the orbital and co-rotation velocities as our a-priori relative velocity: v = v + v r,i=0 o c (6.43) The possibility of including modelled in-track and vertical wind velocities in the algorithm computation will be discussed at the end of this Section. While modifying the direction of the velocity and modelled acceleration vectors, the magnitude of the acceleration is not of importance. We therefore make use of the unit a a a vectors ˆ obs and ˆ mod. Since, according to Equation (6.36), the direction of mod is identi- C cal to the direction of a, their unit vectors will be identical as well, and we can just use the unit vector of the latter to simplify the computation. We can now deﬁne our measure of the acceleration direction residual: d = a − a = a − Cˆ (v ) ˆ obs ˆ mod,i ˆ obs a,i r,i,... (6.44)

In practice, if the magnitude of d is below a certain predefined threshold , convergence has been reached. Otherwise, another iteration is required. The convergence criterion is thus: ||d|| < (6.45) The unit vector representing the direction of the velocity adjustment for the current iteration is defined to be perpendicular to both the relative velocity and the rotation axis: v × e v = r,i ûp ˆ adj,i ||v × e || (6.46) r,i ûp

Next, to start our numerical differentiation, two relative velocity vectors are formed, which keep the magnitude of the unadjusted relative velocity, but which are rotated slightly in both directions with respect to the relative velocity of the current iteration: v + δv v − δv v+ = ||v || r,i ˆ adj,i v− = ||v || r,i ˆ adj,i r r,i ||v + δv ||, r r,i ||v − δv || (6.47) r,i ˆ adj,i r,i ˆ adj,i

78 These modiﬁed relative velocities will result in modiﬁed modelled acceleration directions. The result from both rotation directions is substituted into Equation (6.44): d+ = a − Cˆ (v+ ) d− = a − Cˆ (v− ) ˆ obs a r ,... , ˆ obs a r ,... (6.48) The vector difference between the two velocity vectors is: Δv = v+ − v− r r r , (6.49) and the effect of this velocity rotation on the acceleration direction residual is: + − Δd = ||d || − ||d || (6.50)

Now, all the elements are in place to compute the next iteration of the relative velocity, which keeps the magnitude of the original velocity, but changes the direction. v − (Δv Δ ) v = ||v || r,i d r/ d r,i+1 r,i ||v − (Δv Δ )|| (6.51) r,i d r/ d At this point Equations (6.44) and (6.45) are reevaluated. If the convergence criterion of Equation (6.45) is met, we can proceed computing the crosswind speed and mass density.

v = v − v w,cr r,i r,i=0 (6.52)

2m ||a || ρ = obs (6.53) 2 ||C || Arefvr,i a,i

Modelling of in-track and vertical winds In the description of the algorithms above, we have not discussed the possible effect on the aerodynamics of wind components other than the cross-track component. Since v we are interested in retrieving the crosswind w,cr from the accelerometer data, a model value for this component should not be included in the a-priori relative velocity of Equa- v tion (6.43). A model value for the in-track wind w,it, and the wind in the direction of the v rotation axis w,z could be applied in that equation however. These can be computed by projecting the full model wind on the unit vectors in these directions. v =(v · v ) v w,it w,mod ˆ r ˆ r (6.54) v =(v · e ) e w,up w,mod ˆup ˆup (6.55) v Since ˆ r changes its direction during the iterative process described in the previous Sec- v tion, Equation (6.54) will have to be reevaluated and r in Equations (6.47) and (6.51) will have to be updated after each iteration step.

6.6 Data editing and ﬂagging

The data editing and ﬂagging is currently implemented as follows: • Visual inspection of the input, output and intermediate data (most notably Euler angles, common-mode accelerations, ion engine activation, density and crosswind);

79 • Recording of time intervals containing suspicious data in a text ﬁle;

• This text ﬁle can be accessed during data export, to prevent exporting of suspicious data, or results based on suspicious data;

The ﬁrst step in this process will be partly automatated in a future version of the processing tools. It is planned that certain criteria are checked, such as valid ranges for the various data ﬁelds.

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