# LEO- and Gravity Field Determination

One of many important applications of Global Navigation Satellite Systems (GNSS) is the precise orbit determination (POD) of Low Earth Orbiting (LEO) satellites, i.e., satellites flying at low altitudes of roughly 200-2000 km. Many of these satellites carry on-board GNSS (up to now mainly GPS) receivers which allow the continuous tracking of GPS satellites and the absolute positioning of the GPS antenna with cm accuracy. Such accuracies are mandatory, e.g., for altimetry missions, where radial orbit errors directly degrade the height measurements.
We distinguish between three different types of orbits:

• Kinematic orbit. This is a collection of three dimensional satellite positions at discrete measurement epochs. The positions are purely geometric solutions of a kinematic positioning and are thus fully independent of any force models which govern the motion of the LEO. A kinematic orbit does not yield information on the satellite position between the measurement epochs nor on the satellite velocity. Since kinematic orbits are fully independent of the underlying force models of the LEO satellite, they are well suited for the recovery of the long-wavelength part of the Earth’s gravity field.
• Dynamic orbit. These orbits are particular solutions of the satellite’s equation of motion, which, in the best case, includes all possible accelerations acting on the satellite. In the dynamic orbit determination the GPS observations are used to estimate the initial conditions (position and velocity at some epoch) or, equivalently, the six Keplerian elements together with additional parameters entering the equation of motion. Dynamic orbits fully depend on the underlying force models and their quality is therefore heavily depending on the quality of these models. This type of orbit yields the position and velocity of the satellite at any desired epoch.
• Reduced-dynamic orbit. These orbits again satisfy the satellite’s equation of motion which is defined by the force models. But the strength of the force models is, to some extent, reduced by introducing so-called pseudo-stochastic orbit parameters, like instantaneous orbital velocity changes (pseudo-stochastic pulses) or piecewise constant accelerations. These parameters are unphysical but are a very efficient tool to absorb any un- or mismodeled acceleration acting on the satellite (like, e.g., air drag or solar radiation pressure). Reduced-dynamic orbit determination is well suited to compute LEO orbits of highest quality.

Kinematic LEO orbit positions can serve as pseudo-observations for a subsequent gravity field determination. At the AIUB this is realized in the framework of the Celestial Mechanics Approach (CMA) as a common orbit and gravity field estimation. Not only the parameters of a (reduced-)dynamic orbit, but also the gravity field part of the underlying force models (e.g., the coefficients of a spherical-harmonic expansion of the gravity field) are estimated simultaneously. There are a number of dedicated gravity missions the data of which were and are successfully processed at the AIUB to produce high-quality gravity fields:

• The CHAllenging Minisatellite Payload (CHAMP) satellite was the first satellite to measure the gravity field based on GPS observations.
• The Gravity Recovery And Climate Experiment (GRACE) twin satellites deliver, apart from the GPS-based absolute positions, ultra-precise inter-satellite range measurements. These measurements serve as additional, very strong observations for the gravity field recovery and allow for the determination of the high-frequency and time-variable part of the Earth’s gravity field. The same principle was used for the design of the Gravity Recovery and Interior Laboratory (GRAIL) mission, which allowed for the determination of the lunar gravity field with an unprecedented accuracy for both the near- and the far-side of the Moon.
• The Gravity and steady-state Ocean Circulation Explorer (GOCE) had a gradiometer on board, i.e., three pairs of ultra-sensitive accelerometers arranged in three dimensions to directly measure the gravity field tensor. Similar to the other dedicated gravity missions, the long-wavelength part of the gravity field was mainly determined from the GPS-based kinematic LEO positions.

The satellite-based determination of the global gravity field of the Earth (or other celestial bodies) has become an invaluable tool for geophysical research and the monitoring of the complex system Earth.