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## Über dieses Buch

This book on space geodesy presents pioneering geometrical approaches in the modelling of satellite orbits and gravity field of the Earth, based on the gravity field missions CHAMP, GRACE and GOCE in the LEO orbit. Geometrical approach is also extended to precise positioning in space using multi-GNSS constellations and space geodesy techniques in the realization of the terrestrial and celestial reference frame of the Earth. This book addresses major new developments that were taking place in space geodesy in the last decade, namely the availability of GPS receivers onboard LEO satellites, the multitude of the new GNSS satellite navigation systems, the huge improvement in the accuracy of satellite clocks and the revolution in the determination of the Earth's gravity field with dedicated satellite missions.

## Inhaltsverzeichnis

### Chapter 1. The First Geometric POD of LEO Satellites—A Piece of History

The very first precise geometric (i.e., kinematic) orbit determination of a LEO satellite was presented in Švehla and Rothacher (2002), where for the first time double-difference ambiguity resolution was demonstrated using the CHAMP satellite in LEO orbit and the ground IGS network. In Švehla and Rothacher (2003a, b) and later in Švehla and Rothacher (2005a, b) geometric precise orbit determination (POD) was demonstrated to cm-level accuracy and presented as an established technique and as very attractive for gravity field determination. Here we give a chronological overview of the development of the method.

Drazen Svehla

### Chapter 2. Reference Frame from the Combination of a LEO Satellite with GPS Constellation and Ground Network of GPS Stations

In this section we demonstrate the combination of a LEO satellite with the satellites of the GPS constellation and the ground networks of space geodesy techniques (GPS, SLR, DORIS) in the generation of reference frame parameters. We show clear improvements in terrestrial reference frame parameters after the combination of the GPS constellation in MEO with spaceborne GPS, DORIS and SLR measurements from the Jason-2 satellite in LEO orbit, including station coordinates, tropospheric zenith delays, Earth rotation parameters, geocenter coordinates and GPS satellite orbit and high-rate clock parameters. We analyze the impact of the LEO data on the terrestrial reference frame parameters and possible improvements they could bring. (See also (Svehla et al. 2010b).) This is a continuation of the work performed with the GPS data from the Jason-1 satellite, where the strong impact of the LEO data on the global parameters has already been demonstrated by means of simulated GPS measurements and variance-covariance analysis (Švehla and Rothacher 2006a).Terrestrial reference frames are usually defined by a set of station coordinates that are estimated over a long period of time using a combination of different space geodesy techniques. However, in the case of Precise Point Positioning (PPP) of a GPS receiver on the ground or kinematic or dynamic POD of LEO satellites using GPS, reference stations on the ground are not directly used to estimate the orbit of a LEO satellite or coordinates of a GPS receiver on the ground. The PPP of a ground station or POD of LEO satellites is based on an intermediate reference frame defined by the GPS satellite orbits and epoch-wise estimates of GPS satellite clocks. Any error in the GPS satellite orbits and clocks, or in this intermediate space-based reference frame (that is highly temporal in nature), will map directly into the LEO kinematic/dynamic orbit and gravity field determination (CHAMP, GRACE, GOCE), altimetry results (Jason-2, Sentinel-3, etc.) or coordinates of a ground station. Therefore, an instantaneous terrestrial reference frame can be defined as a frame created by the epoch-wise solution of GNSS orbit and clock parameters supported by other space geodesy techniques such as SLR, DORIS and VLBI. In the next section we introduce the concept of phase clocks in order to consistently bridge the gap between ground-based and space-based terrestrial frames and show how a terrestrial frame can be transferred to the LEO orbit avoiding biases associated with the code GPS measurements.At the end we give an insight into the generation of an instantaneous reference frame from different GPS frame solutions (e.g., provided by IGS ACs) by means of least-squares collocation using a so-called intermediate reference sphere in LEO or GNSS orbit. The use of a simple weighted average, which is often used in the combination of GNSS solutions from different IGS ACs without taking into account correlations in time (and space) of each individual solution, will always introduce systematic effects that are not equally distributed over an imaginary sphere at the GNSS orbit height.

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### Chapter 3. Geometrical Model of the Earth’s Geocenter Based on Temporal Gravity Field Maps

We demonstrate that GRACE gravity field maps could be used to derive annual amplitudes and secular rates in the geocenter z-coordinate from the low-degree odd coefficients (“pear-shaped”), i.e., from the C30, C50 and C70, although degree 1 gravity field coefficients are not estimated. This is because “pear-shaped” coefficients are not symmetrical with the equator like even zonnals C20, C40 and C60, and they are big enough relative to other low-degree “pear-shaped” coefficients to absorb any translation rate present when degree 1 gravity field coefficients are not estimated. If degree 1 gravity field coefficients are derived together with all other gravity field coefficients, degree 1 absorbs systematic effects associated to space geodesy techniques and reference frame realization. Therefore, when degree 1 coefficients are not estimated, any rate in the geocenter z-coordinate is reflected in the translation of the “pear-shaped” harmonics. This also follows from the translation of spherical harmonics. We derived the secular rate and annual amplitudes in geocenter z-coordinate from the low-degree odd coefficients (“pear-shaped”) over the last 10 years (GRACE RL05) and compared it with results from the global GPS and SLR solutions, tide-gauge records over the last 100 years and the limited data set of geocenter z-coordinates estimated from the combined orbit determination for the Jason-2 satellite and the GPS constellation. We confirm the initial assumption that temporal gravity field maps provided by the GRACE mission contain an information on the geocenter z-coordinates and estimated annual amplitudes are very close to results from GPS/SLR/LEO solutions. In addition, this approach reveals an interesting information that the asymmetrical mean sea lever rise between the Northern and the Southern hemispheres could be reflected in the rate of asymmetric surface spherical harmonics (“pear-shaped”). Following (Cazenave and Llovel 2010), satellite altimetry observations suggest that the mean sea level has been rising faster over the Southern than over the Northern Hemisphere, whereas recently (Wöppelmann et al. 2014) using selected tide-gauges measurements corrected with the glacial isostatic adjustment (GIA) and GPS velocities report the opposite sign, i.e. the mean sea level rise of 2.0 ± 0.2 mm/yr for the Northern hemisphere and 1.1 ± 0.2 mm/yr for the Southern hemisphere. Based on the 10 years of GRACE gravity field models (GRACE RL05), we can draw the conclusion that difference in the mean sea level rise between the Northern and the Southern hemispheres is reflected in the rate of the z-coordinate of the geocenter and that the mean sea level has been rising faster over the Southern than over the Northern hemisphere (confirmed Church priv. com.). At the end we derive similar approach from the rates in the even degree zonal spherical harmonics and derive a rate in the scale of GRACE gravity fields of -0.5 ppb/10 yr. This shows that GRACE gravity field maps represented by spherical harmonics contain a scale and one can use temporal gravity field maps to monitor its variations over time.

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### Chapter 4. First Phase Clocks and Frequency Transfer

In Švehla and Rothacher (2004a, b, 2005a, 2006b), it was demonstrated for the first time that clock parameters for GPS satellites and ground stations can be estimated solely from the carrier-phase GPS measurements. These also allow frequency transfer with a very high level of accuracy of a few parts in 10- 16 (≈25 ps/day in terms of linear time rate). The main motivation for the development of the phase clock approach is to avoid the colored systematic noise that is introduced by using code, or smoothed code GPS measurements and other possible biases in the official GNSS clock parameters provided by IGS. On the other hand, phase clocks completely absorb the GPS radial orbit error and are fully consistent with the LEO carrier-phase measurements when determining kinematic or reduced-dynamic LEO orbits, since in both cases carrier-phase ambiguities are estimated. Phase measurements from a GPS ground network of about 40–50 stations tracking about 30 GPS satellites in MEO orbit form a closed, internally connected system, in which the phase information of one clock can be related to that of any other GPS satellite or a ground station clock in the network, even on the antipodal side of the world. This opens up the possibility of high-precision positioning and especially intercontinental non-common view frequency transfer of utmost accuracy. We may say, phase clocks are the optimal way to compare phase information between ground station clocks and/or LEO/GNSS satellites. Later on in this thesis, we introduce the concept of track-to-track ambiguities to optimally fix carrier-phase ambiguities to their integer values.Later, phase clocks were also studied in Dach et al. (2005, 2006); Bauch et al. (2006) and in Matsakis et al. (2006) over longer periods of time and have been compared to other time/frequency comparison techniques. Ambiguity resolution with phase clocks was demonstrated for the first time in Švehla and Rothacher (2006a) and later on in Mercier and Laurichesse (2007), Delporte et al. (2007, 2008). Starting with GPS Week 1449, JPL started providing additional information on clock time bias and drift relative to the reference clock in the IGS network in their IGS reports, see Desai (2007). In their IGS reports, as a reference clock JPL uses exclusively IGS station USN3 (US Naval Observatory), or in some cases AMC2 (Colorado Springs). Besides CNES, all IGS Analysis Centers provide satellite clock parameters calculated using carrier-phase and pseudo-range measurements in order to support both time and frequency transfer at the same time. Thus, IGS clock parameters are more applicable to PPP (Precise Point Positioning) than to frequency transfer. This section describes the estimation of phase clocks and their application in frequency transfer and precise point positioning.

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### Chapter 5. First Geometric POD of GPS and Galileo Satellites

We have already estimated purely geometric orbits of several LEO satellites, and now one may ask how accurately a GPS satellite orbit can be estimated purely geometrically, i.e., kinematically. The main problem is that GNSS satellites are high above the Earth and positioning geometry is not as good as for satellites in LEO orbit. This section deals with the first estimation of one GPS satellite fully geometrically. New Galileo satellites are equipped with H-masers and in this case the satellite clock can be modeled very efficiently using a linear model over one day. We present here the first Galileo orbits estimated geometrically using a linear model for the H-maser on board the GIOVE-B satellite. The current accuracy of geometric GPS orbits is approximately 15 cm, whereas this improves to several centimeters in the case of Galileo. On the other hand, with Galileo, ambiguity resolution on the zero-difference level will be significantly improved, thus once the phase ambiguities are fixed, it is assumed that it will be feasible to estimate GNSS orbits fully geometrically with an accuracy comparable to dynamic orbits. For more on geometric POD of GNSS satellites see (Švehla and Rothacher 2005).

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### Chapter 6. Kinematics of IGS Stations

For comparison with the kinematic POD of LEO and GPS satellites, a ground GPS baseline from Greenbelt (GODE, US) to Algonquin Park (ALGO, Canada) with a length of 777 km was processed kinematically for a period of one day. The coordinates of one station of the baseline were kept fixed (GODE) and a set of three coordinates was estimated every 30 s for the second station ALGO using carrier-phase data only.

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### Chapter 7. Reduced-Kinematic POD

Here we present the results of reduced-kinematic POD, as introduced and published in Švehla and Rothacher (2005). Reduced-kinematic POD can be defined as the fourth fundamental approach in precise orbit determination, along with kinematic, reduced-dynamic and dynamic POD. The main difference between reduced-kinematic and reduced-dynamic orbit determination is that in the reduced-kinematic POD the constrained normal equations are set up for the epoch-wise kinematic positions (with epoch-wise clock parameters), whereas in the reduced-dynamic approach, dynamic parameters (such as initial Keplerian state vector, aerodynamic drag coefficients, empirical accelerations, etc.) and/or some pseudo-stochastic parameters are determined. Thus, in the case of reduced-kinematic POD, degrees of freedom are reduced towards a dynamic orbit, whereas in the reduced-dynamic orbit, the dynamics of the orbit is reduced towards a kinematic orbit. Due to the relative or absolute constraints that are used in the reduced-kinematic POD, we did not use nor develop this approach further for LEO satellites. We merely present typical results for the sake of completeness.

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### Chapter 8. First GPS Baseline in Space—The GRACE Mission

In Švehla and Rothacher (2004) it was reported for the first time that the orbit vector between the two GRACE satellites equipped with GPS in the LEO orbit can be estimated with mm-level accuracy. This level of accuracy was achieved after performing ambiguity resolution for the GPS double-difference baseline and independently confirmed by the K-band measurements between the two GRACE satellites. Here we present the results of this GPS baseline in space.

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### Chapter 9. Geometrical Modeling of the Ionosphere and the Troposphere with LEO Orbit

In this section, we first briefly describe the mathematical and physical background of the first and second order ionosphere effects on LEO GPS measurements and then give a geometrical interpretation of the second order ionosphere effect for one-way and two-way LEO tracking observables. We discuss systematic effects resulting from higher order ionosphere effects on LEO orbit determination and then on gravity field and altimetry results. We show that, when the IGS TEC maps are compared to the TEC observed along the CHAMP orbit (merely by applying a constant bias) during the solar maximum, the agreement is excellent and is at the level of about 1 TECU or below. We show how to calculate the fractional TEC below or above the LEO orbit, taking into account the Sun’s position w.r.t. LEO orbit. We show that the fractional TEC for LEO orbit can be calculated exactly from the Chapman function, by transforming the Chapman function into the “error function” erf(x), encountered when integrating the normal distribution in statistics. This allows a direct combination of LEO and ground IGS TEC maps. After that, we present a novel remove-restore approach in the combination of LEO and ground-based TEC measurements by means of least-squares collocation. The same approach could be applied to augment final and real-time IGS TEC maps. It is proposed to model the ionospheric TEC (by combining LEO and ground GNSS measurements) as a spherically-layered electron density distribution in three main Chapman layers, i.e., E, F1 and F2 with an additional layer for the plasmaspheric density above the ionosphere, using GOCE (above the E-layer), GRACE (above the F1 layer) and Jason-2 (above the F2-layer and below the plasmasphere). In the second part, we discuss tropospheric effects on the propagation of microwave and optical measurements and show the influence of tropospheric effects on the kinematic and reduced-dynamic POD of LEO satellites. We show that there is an effect of the tropospheric modeling on the estimated low-degree zonal gravity field coefficients based on LEO orbits. At the end, we propose a way forward in modeling ground-specific high-resolution tropospheric delays for all space geodesy techniques, making use of the high-performing clocks on board the new GNSS satellites and the more than 35 GNSS satellites in the field of view of a ground station, given that all four GNSS constellations will be deployed in a few years from now. For that, ground-specific tropospheric and ionospheric delays could be modeled making use of the rotation of spherical harmonics in order to account for temporal variations w.r.t. a fixed frame. Rotations of spherical harmonic coefficients provide continuous TEC information.

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### Chapter 10. Aerodynamics in Low LEO: A Novel Approach to Modeling Air Density Based on IGS TEC Maps

Here we present some theoretical aspects of the modeling of aerodynamic acceleration in the precise orbit determination of a LEO satellite. We have included this section because of the great importance of the role that aerodynamic drag plays in all gravity field missions, as they are typically placed in a very low LEO orbit. Thus, here we look at the geometrical properties of this effect. We show that the accuracy of the velocity in the calculation of the aerodynamic drag for a LEO satellite, in particular the velocity of thermospheric horizontal winds, is as important as the atmospheric density. We then give a geographical representation of the models used to calculate atmospheric density and thermospheric horizontal winds, with an emphasis on the GOCE (Sun-synchronous) orbit, and compare this with the orbits of altimetry satellites in high LEO. In addition, we present the prospects of investigating atmospheric density and thermospheric winds using the GOCE mission at 220–250 km altitude. Models of neutral horizontal winds show that thermospheric winds mainly occur around the geomagnetic poles where they are driven by the perturbations in the geomagnetic field. The highest thermospheric wind velocities may be expected along the dawn-dusk regions, and from that point of view, the GOCE orbit is the perfect candidate to provide unique information on the neutral horizontal winds in the lower thermosphere. Section 10.3 of this thesis triggered an ESA study that demonstrated the retrieval of thermospheric wind parameters from GOCE data. At the end of this section, we demonstrate a novel approach to calculating and predicting air density in the thermosphere based on the global TEC maps provided by IGS. This approach could be used to predict solar activity in an alternative way, independent of the number of Sun spots or the solar flux index at a wavelength of 10.7 cm (F10.7). We also show that information on the ionization of the thermospheric part of the ionosphere, as provided in IGS TEC maps, can be used to calculate the LEO mission duration (as was done for GOCE). This opens up new applications for the global IGS TEC maps in monitoring air density in the thermosphere, including spatial and temporal variations. In addition, we show that variations in air density driven by variations in solar activity (heating) are empirically proportional to the ionization of the ionosphere. Thermospheric density and TEC can be related by an empirical linear model as shown here.

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### Chapter 11. GPS Single-Frequency: From First cm-POD to Single Frequency GNSS-RO/R

In this section we introduce what we call “Positive Code-Phase” linear combination or the LP linear combination (phase and code added) to eliminate the first-order ionosphere effect and estimate LEO orbits using single-frequency GPS measurements, (see Švehla and Rothacher 2003a; 2005b). We do not smooth code measurements with the linear model as proposed by the GRAPHIC (Group and Phase Ionospheric Calibration) linear combination in (Yunck 1993; Gold et al. 1994; Muellerschoen et al. 2004). We show that in the case of the GRACE-B satellite it is possible to estimate LEO orbits to an accuracy of 2–3 cm RMS (1.3 cm radial) using single-frequency GPS measurements only, (see also Svehla et al. 2010a). This is similar to the orbit accuracy of 1–2 cm one can typically achieve with dual-frequency carrier-phase measurements. This is possible due to the very low noise level of the code measurements from the GRACE-B satellite and recent gravity field models from the GRACE and GOCE missions that provide very accurate gravity field coefficients up to degree and order 120 allowing an orbit parameterization with a very modest number of empirical parameters. In addition, thanks to the excellent precision of the real-time GPS satellite clock parameters provided by the IGS, we show that this cm-orbit accuracy can be achieved even in real-time. Subsequently, we introduce an estimation of the group delay pattern of GNSS satellite antennae based on the LP linear combination. We show that the LP linear combination can be used to estimate single-code group delay variations (GDV) for GNSS satellite antennae at the single-frequency level and present the first GDV pattern based on GPS measurements from the GRACE-B satellite. The GDV pattern based on LP linear combination is related to a single code observable and not to an ionosphere-free linear combination, a strong advantage in the presence of multi-GNSS data. After that, we present the concept of using single-frequency GPS radio-occultations (RO) as a very promising alternative to standard GPS-RO based on dual-frequency measurements. The advantage of this approach is that carrier and code measurements on the same GPS frequency follow the same path in the ionosphere. This is not the case for the bended carrier-phase GPS-RO measurements on different GPS frequencies that can reach a vertical separation of up to 500 m in some cases. Since the antenna used for GPS-RO is typically a high-gain antenna, the noise level of the code measurements is very low and, with an additional smoothing, this approach could be used for GPS-RO with SBAS satellites in GEO. The same approach could also be applied to GNSS reflectometry (GNSS-R).

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### Chapter 12. Absolute Code Biases Based on the Ambiguity-Free Linear Combination—DCBs Without TEC

Absolute code biases and associated DCBs determined using absolute code biases are called “absolute” because they do not require TEC information to estimate them and are defined against the IGS Clock Convention (“P3 clocks”).Differential code biases (DCBs) are typically determined by co-estimating the first-order ionosphere effect using the geometry-free linear combination of code measurements from two different GNSS frequencies. We develop ambiguity-free linear combinations based on the dual- or triple-frequency GPS carrier-phase and code measurements on only one GPS frequency. In this way, we can estimate code biases on a single GPS frequency. Since the datum of the GPS satellite clock corrections is defined by the ionosphere-free linear combination of the P-code measurements on L1 and L2 we can estimate these single-frequency code biases as “absolute biases” using the geometry-free approach. Our ambiguity-free linear combination removes single-frequency ambiguities, but it requires the estimation of one wide-lane ambiguity with a very long wavelength, a wavelength that is significantly greater than the size of the code biases. In addition, by forming single-differences between two GNSS satellites using measurements from one station, one can separate satellite-based from station-based code biases. We show the relationship between the code biases and the narrow-lane biases in the Melbourne-Wübbena linear combination and DCBs. The same approach is extended to other multi-GNSS code observables.Absolute code biases defined for single-frequency observables can be used to combine carrier-phase and code measurements consistently in a multi-GNSS environment and to define carrier-phase ambiguities and ionospheric effects in an “absolute sense”. Absolute code biases can provide a datum for estimated global ionosphere maps and for all calibration of multi-GNSS code measurements (e.g., group delays). We show here absolute code bias in P1 and C5 code GPS measurements on L1 and L2 carrier-phases and present calibration of ¼-ambiguities associated with L5. We discuss absolute code biases in the light of the S-curve bias and group delay variation maps for GNSS satellites. We show how, by introducing absolute code biases, we can consistently define a datum for GNSS satellite clock parameters and ionosphere maps in a multi-frequency GNSS environment. Galileo and future GNSS will introduce wide-band signals that will lead to low code noise (in the cm-range). Specifically, the Galileo E5 wide-band signal (nominal bandwidth of 51.15 MHz) and the AltBOC modulation will offer code noise at cm-level. The same approach could be applied to Galileo using wide-band signals as reference signals to determined absolute code biases.

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### Chapter 13. LEO Near-Field Multipath and Antenna Effects

In an internal technical note (Švehla and Rothacher 2004b), it was suggested to the GOCE Project Office in ESA that a study be conducted on the effect of the near-field multipath on a POD antenna due to the structural environment of the GOCE satellite itself. The idea was that by performing an absolute calibration of the GOCE antenna, with and without a mock-up (solar panel wing), the near-field multipath effect could be described as the difference between the two estimated PCV maps. In the case of near-field multipath, the total antenna PCV correction can be defined as the sum of the nominal antenna PCV map and the antenna map resulting from the near-field multipath. This section studies multipath effects originating from the satellite environment and the impact of GPS antenna calibration on orbit determination of LEO satellites. It is shown that near-field multipath has a very strong effect on the kinematic POD of a LEO satellite using carrier-phase measurements. At the end of this section, a near-field multipath calibration method is proposed and then discussed for GNSS satellites.

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### Chapter 14. Probing the Flyby Anomaly Using Kinematic POD—Exotic Applications of Kinematic POD

The idea presented here is to use the GPS receiver for the comparison of kinematic and dynamic orbits of an interplanetary mission during Earth flyby, e.g., BepiColombo, Juno. Purely geometrical orbits can be estimated to an accuracy of 1 cm RMS using GPS carrier-phase measurements, whereas dynamic orbits will be affected by any potential flyby anomaly effect on the spacecraft while it is in Earth flyby.

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### Chapter 15. Galileo-2: A Highly Accurate Dynamical GEO Reference Frame to Complement the TRF

In Švehla (2007), Švehla et al. (2008) and in Švehla (2008) we presented a novel design for the GNSS system called here Galielo-2 based on recent developments in optical clocks, frequency combs and time/frequency comparison technology. We demonstrated a concept of a navigation system in MEO based on master clocks in the GEO orbit and two-way optical/microwave links to transfer their stable frequency to the navigation satellites in MEO orbit (either from the ground or via GEO). In this way, the use of H-masers and Cs- or Rb-clocks in the GNSS satellites can be avoided and frequency combs could be used to generate the desired navigation radio (and optical) signal in the MEO orbit. The development of ˝Ultra˝-USO, e.g., for the STE-QUEST mission with a frequency stability in the order of 10-15 at 1 s is sufficient to meet the required GNSS clock stability over a longer period of time (e.g., one day), and thus one could separate precise orbit determination of GNSS satellites from estimation of GNSS clock parameters. GNSS clock frequency can be steered either from the ground or from the GEO orbit making use of the two-way metrology links. For this, master clocks in GEO do not need to be of the highest accuracy, they could be optical clocks or the latest Rb-clocks with high short-term stability. However, the assembly of several GEO clocks equipped with optical/microwave links for frequency transfer will meet the needs of the timing community for clock comparison in the generation of the global TAI/UTC time scale. Thus, the idea of Galileo-2 is twofold: Firstly, to combine positioning and timing systems under one umbrella, and, secondly, to enable new applications in geosciences.Generally speaking, a highly accurate dynamic reference frame in the GEO orbit would, in future, have the potential in terms of accuracy to provide an alternative to, and to complement, the terrestrial reference frame of the Earth. Drag-free and ranging technology as developed for the LISA mission provide very strong arguments in this direction. A GEO reference frame could provide the basis for a real-time positioning/timing facility for all GNSS Earth-based applications, from LEO to GEO orbit and beyond towards lunar orbits. Intersatellite ranging between such (drag-free) GEO satellites could be obtained to a very high level of accuracy, e.g., sub-micrometer — several orders of magnitude higher than the accuracy of a terrestrial reference frame. Considering the orbit-redshift equivalence principle we introduce in Sect. 29 (a symmetry between the error in orbit position and velocity such that these cancel or compensate each other out in generating the net redshift effect), an orbit in space (GEO) offers the best environment to define and realize the frequency standard and define the SI second using an atomic clock. A far more reliable method than using the geoid and the surface of the Earth. This is mainly due to the fact that cold atoms in the clock can be observed for a long time in space (weightlessness) and are not limited by free-fall as they are on Earth. This typically gains an additional 3–4 orders of magnitude in sensitivity. Therefore, in future, GEO orbit could offer the best place to define the datum for time and so support positioning on Earth. The terrestrial reference frame of the Earth is, by definition, tied to the ground network of station coordinates on the Earth’s crust. Thus the proposed realization using GEO orbit is an extended and complementary realization of the terrestrial frame which aims to achieve higher accuracy and precision and to obtain synergy with time realization.

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### Chapter 16. The GPS Transponder Concept—Towards One-Way and Two-Way GNSS Frequency Transfer

In this section we discuss alternative, geometry-free approaches for positioning and time/frequency transfer using one-way and two-way measurements. The transmitter and receiver clock parameters can be separated or removed from the tracking geometry by using two-way measurements or introducing one-way measurements into the geometry-free linear combination. Clocks on board GNSS have become so stable that it makes interesting to steer their frequency using a geometry-free approach as demonstrated here. Galileo satellite clock parameters can be modelled using just two parameters per day (time drift and offset) with the remaining residual clock parameters showing the standard deviation at the level of 15 mm, see Sect. 18. Therefore, frequency steering of the satellite clock could be performed far more infrequently, (e.g., once a day) using the two-way frequency transfer approach. This could also bring to the separation of the prediction of GNSS satellite clock parameters (based on frequency steering) from the orbit prediction. We also discuss an application of the one-way frequency transfer approach based on geometry-free linear combination between two satellites (e.g., between GNSS satellites in MEO or with GEO). On the development of the two-way microwave metrology links for atomic clocks of the ACES mission we refer to Cacciapuoti and Salomon (2009).In addition to providing a two-way frequency transfer capability for GNSS, one could also consider the GPS-transponder concept, where a GNSS signals is tracked by a LEO GNSS receiver and then re-transmitted by the LEO satellite to a ground station (e.g., on a slightly shifted frequency). This opens up the possibility of separating tracking geometry from clock information when using a one-way approach for positioning, similar to the geometry-free two-way approach. One could also consider combining the standard one-way GPS positioning with the one-way frequency transfer. Observables in the one-way frequency transfer based on geometry-free linear combination would then be free of propagation effects, such as the effects of the ionosphere and the troposphere. The one-way approach based on geometry-free linear combination would also eliminate errors due to tropospheric effects and atmospheric turbulence in the case of optical measurements, and tropospheric effects and first and higher-order ionospheric corrections in the case of microwave measurements.We also discuss the geometrical mapping of GNSS constellations with VLBI against extragalactic radio sources in the GPS-transponder configuration. At the end of this section, we discuss the idea of a similar two-way approach constructed using VLBI to observe both LAGEOS and passive laser retro-reflectors on the Moon in a bi-static radar configuration.

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### Chapter 17. The SLR/LLR Double-Difference Baseline

Here we present a novel SLR double-difference approach with GNSS satellites. It is shown how forming double-differences of SLR measurements between Herstmonceux (HERL) and Graz (GRZL) ILRS stations and two Galileo satellites removes common SLR biases: i.e., ILRS station range biases and common retro-reflector effects. By using the orbits of GNSS satellites from IGS as fixed in the parameter estimation, the double-difference SLR approach offers a bias-free estimation of relative coordinates with the mm-accuracy between two ILRS stations (SLR baseline) that are separated by about 5000 km. In this way, we obtain SLR observables of utmost precision and accuracy at sub-millimeter level with the standard deviation σ = 0.5 – 1.0 mm. We show that after differencing the remaining noise in the SLR measurements nicely averages out, leading to estimation of station coordinates, local ties between different space geodesy techniques and precise comparison of optical/microwave tropospheric effects. Considering that relative station coordinates between ILRS stations can be estimated in a similar way between collocated GNSS stations using the GNSS double-differences, the SLR approach allows direct estimation of local ties between SLR and GNSS ground stations. We extend the common-view SLR and make double-differences over time by considering the different observation times for all SLR measurements between all SLR stations. SLR range biases and small biases between SLR sessions are removed. The scale is preserved when double-differencing SLR and free of range biases (at mm-level), making this approach very attractive to combine ILRS network with IGS network in the global GNSS solution. We show that LLR offers estimation of UT0 and with differential SLR the global GNSS can estimate a complete terrestrial frame. For the un-differenced SLR we refer to Pearlman et al. (2002).When a LEO satellite is observed by two SLR stations quasi-simultaneously with a GNSS satellite, one can calculate the “vertical SLR baseline” (vector) between the GNSS and the LAGEOS (LEO) satellite as well as the “vertical SLR range” (GNSS-LEO range) derived from geometry. This provides radial orbit information that can be used for altimetry and gravity field missions as well as reference frame satellites. At the end we extend the double-difference approach to other space geodesy techniques such as lunar laser ranging, VLBI and DORIS and discuss estimation of local ties and global reference frame parameters. We also derive a relationship between a possible bias in LAGEOS center of mass correction and radial bias in GNSS orbits. At the end we extend the concept of SLR double-differencing to lunar laser ranging (LLR) and present first results for the LLR double-difference baseline. We succeeded in processing LLR measurements to Apollo and Luna retro-reflectors on the Moon, and, in a similar way, have processed SLR measurements to GPS satellites considering only the geocentric frame in order to model the uplink and downlink for lunar laser ranges.

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### Chapter 20. Track-to-Track Ambiguity Resolution for Zero-Differences—Integer Phase Clocks

In this section we introduce a novel approach for GNSS ambiguity resolution at the zero-difference level, what we call Track-to-Track (T2T) ambiguity resolution. The T2T approach is based on the resolution of wide-lane and narrow-lane ambiguities between consecutive satellite tracking passes, what we call track-to-track or pass-to-pass ambiguities. To fix T2T ambiguities to their integer values, GNSS measurements from only a single GNSS receiver are used without forming any double-differences or similar combinations between different GNSS receivers. Thus, the T2T approach is especially appropriate for LEO applications, to connect very short tracking passes (typically 15–20 min) that introduce a very large number of zero(double)-difference ambiguities, and for ground networks, where the ambiguities of a single GNSS satellite can be connected over a longer period of time (e.g., one week). This opens up a new application for T2T ambiguities to monitor stability and to define code biases and GNSS clock parameters over a long period of time. In this section, we demonstrate the T2T ambiguity resolution approach using LEO and ground GPS measurements. We show that LEO T2T ambiguity resolution leads to an optimal combination of LEO and ground GPS measurements and thus opens doors to form a network of LEO satellites in space for the determination of combined GNSS/LEO terrestrial reference frame parameters. This is possible thanks to the connected LEO ambiguities over all tracking passes (about 16 ambiguities per day per GPS satellite). Hence double-differences between a LEO satellite and ground stations are connected, reducing the number of zero-difference or double-difference ambiguities with the ground IGS network by nearly 95%.The same Track-to-Track (T2T) ambiguity resolution approach based on carrier-phase measurements could be applied to double-differences. Biases in the double-differences that are common and repeat from one GPS tracking pass to another tracking pass (e.g., multipath effects, orbit errors, etc.) will be removed when forming differences of double-difference ambiguities between consecutive tracking passes. This is particularly true for the narrow-lane ambiguities where the reduction of common systematic effects between tracking passes will significantly improve ambiguity resolution. In this way reducing the effects like near-field multipath and orbit errors, that repeat in a similar way from the track to the track.

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### Chapter 21. Integer Ambiguity Algebra

In this section we develop integer ambiguity algebra, a mathematical approach to handle integer ambiguities between different GNSS frequencies and introduce what we call the ambiguity-free linear combination. We first show the vector form of the wide-lane ambiguity for multi-frequency GNSS and then develop integer ambiguity algebra and show in detail the integer property of the ionosphere-free ambiguity for GPS and Galileo. We show that any GNSS ionosphere-free linear combination can be represented by an integer ambiguity without resolving wide-lane ambiguity. This opens up the possibility of forming an integer ambiguity of arbitrary wavelength, when combined with narrow-lane ambiguity. We introduce an elegant way to resolve wide-/narrow-lane ambiguities using the ambiguity-free linear combination that is consistent with what we term absolute code biases. The advantage of this approach is the consistent resolution of wide-lane ambiguities and calibration of wide-lane biases in an absolute sense, since the same ambiguity-free linear combination can be used to estimate absolute code biases, (see section on absolute code biases). Code biases can be defined in an absolute sense if one uses the IGS convention for estimated clock parameters that the net effect of code biases is zero for the ionosphere-free linear combination of P-code measurements, or so-called P3-clocks. They are still limited by the full number of wide-lane ambiguities that can be defined separately for two- and three-carriers with a wavelength of 0.67 m and 3.41 m respectively. Since absolute code biases are determined against the ionosphere-free P-code, we obtain a consistent framework for ambiguity resolution for all four GNSS. Then, by using integer ambiguity algebra, we develop three-carrier wide-/narrow-lane linear combinations for GPS/Galileo and show how to use this approach for ambiguity resolution and retrieval of ionospheric effects. We show that a three-carrier-type Melbourne-Wübbena linear combination can be derived by means of ambiguity algebra.

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### Chapter 22. Earth Orientation Quaternion

In Švehla 2006, it was proposed for the first time to represent Earth orientation and rotation by means of an Earth Orientation Quaternion (EOQ). Quaternions are a very practical way to represent the Earth’s orientation parameters (EOPs), because the transformation between the terrestrial and the inertial system can be performed without calculating rotation matrices. Most importantly, the use of EOPs stored in the form of a quaternion avoids the use of the latest models and standards available from the IERS Conventions, as in the case of the EOP/ERP parameters provided by IGS and IERS. In this way, information about the Earth’s rotation/orientation is straightforward and the transformation can be performed much in the same way as for satellite attitude. This idea that was originally presented in Švehla (2006), was included in the recommendations of the Workshop on Precise Orbit Determination for the future ESA Earth observation missions, held at ESA/ESTEC in 2007 (Švehla 2007c). Following this recommendation, the ESA Core Mission GOCE provides Earth Orientation Quaternions as a separate product accompanying the kinematic and reduced-dynamic orbit. The sampling rate of Earth Orientation Quaternions, as provided in the scope of the GOCE mission.The four Euler symmetric parameters written in the form of a quaternion are a minimal set of parameters for defining non-singular mapping to the corresponding rotation matrix. Besides their symmetrical properties, modeling finite rotations using quaternions has many advantages compared to using Euler angles since any interpolation or integration can be performed on the sphere, preserving the orthonormality of the rotation transformation (Švehla 2006).Hamilton or quaternion algebra avoids the use of a rotation matrix and any sequence of successive rotations can be represented very elegantly by the quaternion operator. This also holds for the derivatives of the successive rotations and the treatment of the kinematic equation of rotation. We show how to interpolate and extrapolate the Earth orientation quaternions preserving the orthonormality of the transformation. We introduce a transition quaternion derived from the kinematic equation of rotation.In the field of numerical solutions of ordinary differential equations, geometric integration is defined as a numerical method that preserves the geometric properties of the exact flow of a differential equation. Therefore, when talking about integrating quaternions on the sphere and preserving orthonormality of the rotation transformation at the same time, we are actually talking about using geometric integration.

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### Chapter 23. A Geometrical Approach to Model Circular Rotations

Here we introduce an elegant way to geometrically model the rotation of a rigid body in vector form. Typically, to perform a rotation in Euclidian space ℜ3 one uses rotation matrices based on a given sequence of Euler angles. Another approach is to use quaternions. A matrix exponent is often used to describe rotations in mathematical expressions and derivations, i.e., the exponential map from so(3) to SO(3). However, the nine elements of the rotation matrix are still exclusively used for calculating rotations in Euclidian space. The axis/angle representation in terms of quaternions and Rodrigues’ rotation formula are alternative approaches. However, hidden geometrical properties, or the complexity of using quaternion algebra are the stumbling blocks that lead to the situation that rotation matrices are still almost exclusively used nowadays. Here we introduce the spherical orthodrome rotation that describes a rotation purely geometrically in a highly transparent way as an orthodrome, or a great arc on a sphere. The application of such transparent geometrical rotations in vector form has many advantages compared to any other rotation. Here we introduce spherical rotation and show basic geometrical properties, i.e., the use of vector algebra to very efficiently perform rotation of a vector in Euclidian space or to describe any orientation. Thus, this approach could be used to model Earth orientation and rotation as well as the attitude of a satellite. We also show that this geometrical rotation approach could be used in orbit modeling, since orbit perturbations can be represented by circular rotations with an axis of rotation very close to the main axis of the satellite orbit.

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### Chapter 24. The Concept of Counter-Rotating Circular Orbits

Here we discuss the concept of bi-circular orbits and bi-circular orbit perturbations. It is shown how an elliptical orbit can be decomposed into two counter-rotating circular orbits. In this way, orbital dynamics can be approximated geometrically by circular orbits or circular rotations. Two counter-rotating orbits remove the variation of the orbit radius. Bi-circular orbit representation is essentially a linear combination of two harmonic oscillators with an opposite direction of rotation. In Chap. 19, we applied a simple harmonic oscillator to daily estimates of residual Galileo clock parameters. We just looked into the remaining amplitude in the clock parameters that measure the radial orbit error after removing a linear model (time offset and drift removed). Similar results to the circular representation of the effect where obtained when a solution of Hill equations in the radial direction (Colombo 1986) was plotted after removing a linear model (bias and drift) in the radial direction, see Chap. 19. The use of harmonic oscillators leads us also to the synergy or unification in modeling of orbital and rotational dynamics. We will show in the next section an interesting feature of circular orbit representation: that for a Keplerian orbit the velocity vector describes a circle. The velocity vector of the satellite in the presence of any point-like mass will rotate about that object along a circle with a constant radius. Thus an interesting application is in supporting numerical integration.Another interesting feature of circular perturbations is in preserving the orthonormality of the rotation transformation, i.e., the geometrical properties of the orbit. The term orthonormality group denotes an orthogonal set of vectors that are normalized in terms of length. Most analytical orbit theories use a form of Keplerian motion as a reference and in numerical integration, typically, higher-order polynomials are used to approximate the orbit over an integration step. Here we use a combination of two uniform circular motions to represent the orbit in terms of orbit positions and in the next section we will see how to use a circular representation and its multipole expansion in modeling orbit velocity.

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### Chapter 25. The Circular Kinematic and Dynamic Equation of a Satellite Orbit

Here we discuss the kinematic equation of a satellite orbit based on a circular representation of the velocity vectors of a Kepler orbit, otherwise known as the two-body problem in celestial mechanics. The velocity vector for Keplerian orbit describes a circle, i.e., we show that the velocity vector of the satellite in the presence of any point-like mass will rotate about that object along a circle with a constant radius. Thus, an interesting advantage of using circular perturbations is that this method preserves the orthonormality of the rotational transformation, i.e., the geometrical properties of the orbit. We show that the proposed circular model could be applied to kinematic as well as dynamic modeling of the orbit and rotation of a rigid body (satellite, Earth, etc.). In the case of circular perturbations, the radius of rotation is preserved, as is also the case with rotation of a rigid body (satellite, planet, etc.). At the end of this section, we discuss the proposed model in the light of geometrical integration, a special kind of integration that preserves the properties of the orbit, i.e., the exact flow of differential equations or Hamiltonian systems that govern satellite motion and rotation. In the light of circular perturbations we extend Newton’s theorem of revolving orbits that defines a special central force as one that is changing the angular speed of the orbit by some constant factor, while the radial motion remains unaffected.

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### Chapter 26. A Geometrical Approach for the Computation and Rotation of Spherical Harmonics and Legendre Functions up to Ultra-High Degree and Order

In this section we introduce a new algorithm for the computation and rotation of spherical harmonics, Legendre polynomials and associated Legendre functions up to ultra-high degree and order. The algorithm is based on the geometric rotation of Legendre polynomials in Hilbert space. It is shown that Legendre polynomials can be calculated using geometrical rotations and can be treated as vectors in the Hilbert space leading to unitary Hermitian rotation matrices with geometric properties. We use the term geometrical rotations because although rotation itself is not governed by gravity and it can be used as a proxy to represent a gravity field geometrically. This novel method allows the calculation of spherical harmonics up to an arbitrary degree and order, i.e., up to degree and order 106 and beyond.

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### Chapter 27. Trigonometric Representations of Legendre Functions

Although the trigonometric representation of associated Legendre functions has been considered in literature, here we give a new insight into the trigonometric reduction of Legendre polynomials. We show that Legnedre polynomials can be calculated up to an ultra-high degree, e.g., n = 106 and beyond without recursive relations and this can be used as a basis for the calculation of associated Legendre functions. The approach presented here was reported for the first time in Švehla (2008) and in Svehla (2010). In addition, we derive orthogonal geometrical forms of associated Legendre functions. However, in terms of performance, our geometrical approach based on the addition theorem of Legendre functions and geometrical rotations along the equator (previous section) is significantly more elegant.

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### Chapter 28. Insight into the Earth’s Interior from Geometrical Rotations in Temporal Gravity Field Maps and Earth’s Rotation

To use the dynamics of GPS satellites to complement geometrical Earth rotation and orientation parameters from VLBI has always been a challenge. Geometric VLBI differs from other space geodesy techniques, such as GNSS, SLR and DORIS, in that it does not rely on satellite dynamics to estimate terrestrial reference frame parameters. Here we present a geometrical approach that combines dynamic and geometric variations in the Earth’s rotation with temporal gravity field variations. We demonstrate that this novel approach provides a new insight into the Earth’s interior, especially into processes and dynamics associated with the Earth’s fluid outer core and the great earthquakes over the last 10 and 100 years. Firstly, we demonstrate that by combining two LAGEOS satellites in low MEO orbit we can remove errors in secular orbit perturbations stemming from low zonal harmonics (J2) and give new insights into the Earth’s rotation and nutation rates. Nutation rates were first estimated from GPS data including orbit determination of GPS satellites (Rothacher et al. 1999). Here we extend the theoretical model of nutation rates and show how, with the nodal separation close to 180° of the two LAGEOS satellites, common orbital errors in terms of nodal and apsidal orbit precession are eliminated. This approach based on celestial mechanics opens up the possibility of using satellite dynamics to determine rates of nutation and variations in length-of-day (LOD) very accurately and correlate them against the variations in the temporal gravity field (errors in J2 are eliminated). This then leads us to the unexplained rate of variations in dynamical LOD estimated from GPS/LAGEOS (orbits driven by Earth’s gravity) and from geometrical LOD from VLBI (external measure of Earth’s orientation). We show how the rotation of spherical harmonics can explain this unresolved effect since rotation of the tri-axial Earth ellipsoid is the real physical phenomenon measured by gravity field missions as well as by SLR to LAGEOS satellites. We show how the geometrical rotation of spherical harmonics is equivalent to temporal gravity field variations and in the case of second degree harmonics is directly proportional to the rate of variations in LOD. This was presented for the first time in Švehla (2008). The conventional IERS mean pole model is in very good agreement with the terrestrial pole of the GRACE monthly gravity field models (derived from $$\bar{C}_{21}$$C¯21̄̄ and $$\bar{S}_{21}$$S¯21 gravity field coefficients). We show that temporal variations in the orientation of the tri-axial Earth ellipsoid (sectorials) are taking place along the equatorial plane, i.e., sharing the same axis of rotation within <0.02 arcsec w.r.t. the IERS mean pole model. This dynamic of the tri-axial Earth ellipsoid is very highly correlated with the major earthquakes over the last 10 (GRACE mission) and 100 years. Recently, Holme and de Viron (2013) showed that variations in the Earth’s rotation that occur with a 5.9-year cycle are probably related to motions within the Earth’s fluid outer core (contemporaneous with geomagnetic jerks). Here we show that temporal gravity field variations in the second degree harmonics, represented by a rotation of the tri-axial Earth ellipsoid, most likely have the definitely same or a similar origin. The idea to study the Earth’s orientation is further extended with a highly elliptical orbit as proposed for the Space-Time Explorer (STE-QUEST) mission in the ESA Cosmic Vision Programme. We discuss the potential of tracking the STE-QUEST satellite in a highly elliptical orbit with VLBI, especially during long apogee dwells, against extragalactic radio sources, thus, combining a geometrical celestial VLBI frame and a terrestrial reference frame. We show how a highly elliptical orbit can be considered as a sensor for Earth rotation, for low-order spherical harmonics coefficients and subsequently for the Earth’s interior dynamics. A satellite dwells for a considerable period of time at the apogee of a highly elliptical orbit, thus it is a perfect target for VLBI to map satellite dynamics against the positions of extra-galactic radio sources. In LEO, a satellite can be observed only for a very short period of time with VLBI and other ground-based techniques. In addition, lunar third-body perturbations are very much uniform along the LEO orbit. Thus, in comparison with HEO, the LEO orbit precesses mainly due to the J2 coefficient of the Earth’s gravity field.

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### Chapter 29. Geometrical Representation of Gravity

To use atomic clocks for the in situ determination of differences in the gravitational potential of the Earth’s gravity field was proposed for the first time by Bjerhammar (1985), Vermeer (1983), taking up Einstein’s postulation that two atomic clocks will tick at different rates due to different gravity potential values at different locations. However, this concept has not been demonstrated so far due to limitations in comparing clock frequency at ≤10−18 relative accuracy between two distant locations. Recently, a frequency transfer was demonstrated below 10-18 relative accuracy over a distance of ca. 920 km using an optical fibre (Predehl et al. 2012), with only one optical clock placed at one end of the optical fibre and a H-maser at the other end. In Švehla and Rothacher (2005b) it was proposed to use atomic clocks in space to measure the gravitational potential along an orbit, to measure together with GNSS, both position and gravity in a purely geometrical way. Here we provide the physical background to relativistic geodesy that is not given in Bjerhammar (1985) and, based on this, provide a geometrical representation of gravity and its relation to orbital motion and reference frames for time. We also show that in special cases, it is possible to measure absolute gravity potential values using quantum mechanics, which opens up new possibilities for the use of state-of-the-art optical clocks. Beyond the Standard Model in theoretical physics based on four fundamental forces, gravitation is still separated from the electromagnetic, strong nuclear, and weak nuclear interactions that are successfully related by the quantum field theory at the level of atomic, particle and high energy physics. On the other hand, general relativity brilliantly describes all observed phenomena related to gravitation in our Solar System and at galactic and cosmological scales. However, general relativity is fundamentally incomplete, because it does not include quantum effects. A unified theory relating all four known interactions will represent a step towards the unification of all fundamental forces of nature. Here we show that circular perturbations could provide an interesting representation between quantum mechanics and orbit mechanics. We try to establish an equivalence between the orbit mechanics based on circular perturbations and basic principles of quantum mechanics. We show that gravity at quantum level and at celestial level can be represented with the same property as light, i.e., gravity and light can be represented as oscillating at the equivalent rate and thus propagate at the same rate. In the essence of every orbit one could consider a wave represented by matter and time that could be modelled or represented by two geometrical rotations. We try to represent gravitational potential by two geometrical counter-rotations, with the rotation of spherical harmonic coefficients as generating functions. This dualistic concept is similar to the electromagnetic force where electricity and magnetism are elements of the same phenomenon orthogonal to each other. Following the general relativity, any form of energy that couples with spacetime creates differential geometrical forms that can describe gravity. Thus, gravitation can be considered purely as a geometrical property. However, our geometrical representation using two counter-oscillations (bi-circular orbits) can be considered as describing gravitation from the scalar point of view at the quantum as well as at the celestial level. Thus it gives geometrical and scalar properties of gravitation at the same time. This is similar to the concept of a magnetic field generated on top of an existing electric field, or similar to the concept of matter and antimatter in particle physics, where antimatter is described as material composed of antiparticles with the same mass as particles, but with opposite charge (leptons, baryons). Following recent results from the Planck mission (Planck Collaboration et al. 2013), there is strong evidence that 26.8% of the mass-energy of the Universe is made of non-baryonic dark matter particles, which should be described by the Standard Model.

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