GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Each orbital plane nominally contains 4 satellites, which are generally not spaced evenly around the ellipse. Therefore, the angle of the satellite within its own orbital plane, the"true anomaly, is only approximately spaced by 90. The true anomaly is measured from the point of closest approach to the Earth(the perigee).(We note here that there are other types of anomaly" in GPS terminology, which are angles that are useful for calculating the satellite coordinates within its orbital plane). Note that instead of specifying the satellites anomaly at every relevant time, we could equivalently specify the time that the satellite had passed perigee, and then compute the satellites future position based on the known laws of motion of the satellite around an ellipse. Finally, the argument of perigee is the angle between the equator and perigee. Since the orbit is nearly circular, this orbital parameter is not well defined and alternative parameterisation schemes are often used Taken together (the eccentricity, semi-major axis, inclination, Right Ascension of the ascending node, the time of perigee passing, and the argument of perigee), these six parameters define the satellite orbit. These parameters are known as Keplerian elements Given the Keplerian elements and the current time, it is possible to calculate the coordinates of the satellite GPS satellites do not move in perfect ellipses, so additional parameters are necessary Nevertheless, GPS does use Kepler's laws to its advantage, and the orbits are described in parameters which are Keplerian in appearance. Additional parameters must be added to account for non-Keplerian behaviour. Even this set of parameters has to be updated by the Control Segment every hour for them to remain sufficiently valid 2.2.2 Orbit design consequences Several consequences of the orbit design can be deduced from the above orbital parameters and Kepler's laws of motion. First of all, the satellite speed can be easily calculated to be approximately 4 km/s relative to Earth's centre. All the GPS satellites orbits are progrado which means the satellites move in the direction of Earth's rotation. Therefore. the relative motion between the satellite and a user on the ground must be less than 4 km/s. Typical values around I km/s can be expected for the relative speed along the line of sight(range The second consequence is the phenomena of "repeating ground tracks"every day. It is straightforward to calculate the time it takes for the satellite to complete one orbital olution. The orbital period is approximately t= ll hr 58 min. Therefore a GPS satellite completes 2 revolutions in 23 hr 56 min. This is intentional, since it equals the sidereal day which is the time it takes for the Earth to rotate 360.(Note that the solar day of 24 hr is not 360, because during the day, the position of the Sun in the sky has changed by 1/365.25 of a day, or 4 min, due to the Earth's orbit around the Sun) Therefore, every day(minus 4 minutes), the satellite appears over the same geographical location on the Earth's surface. The"ground track? "is the locus of points on the Earth's surface that is traced out by a line connecting the satellite to the centre of the Earth. The4 GEOFFREY BLEWITT: BASICS OF THE GPS TECHNIQUE Each orbital plane nominally contains 4 satellites, which are generally not spaced evenly around the ellipse. Therefore, the angle of the satellite within its own orbital plane, the “true anomaly”, is only approximately spaced by 90 o . The true anomaly is measured from the point of closest approach to the Earth (the perigee). (We note here that there are other types of “anomaly” in GPS terminology, which are angles that are useful for calculating the satellite coordinates within its orbital plane). Note that instead of specifying the satellite’s anomaly at every relevant time, we could equivalently specify the time that the satellite had passed perigee, and then compute the satellites future position based on the known laws of motion of the satellite around an ellipse. Finally, the argument of perigee is the angle between the equator and perigee. Since the orbit is nearly circular, this orbital parameter is not well defined, and alternative parameterisation schemes are often used. Taken together (the eccentricity, semi-major axis, inclination, Right Ascension of the ascending node, the time of perigee passing, and the argument of perigee), these six parameters define the satellite orbit. These parameters are known as Keplerian elements. Given the Keplerian elements and the current time, it is possible to calculate the coordinates of the satellite. GPS satellites do not move in perfect ellipses, so additional parameters are necessary. Nevertheless, GPS does use Kepler’s laws to its advantage, and the orbits are described in parameters which are Keplerian in appearance. Additional parameters must be added to account for non-Keplerian behaviour. Even this set of parameters has to be updated by the Control Segment every hour for them to remain sufficiently valid. 2.2.2 Orbit design consequences Several consequences of the orbit design can be deduced from the above orbital parameters, and Kepler’s laws of motion. First of all, the satellite speed can be easily calculated to be approximately 4 km/s relative to Earth’s centre. All the GPS satellites orbits are prograde, which means the satellites move in the direction of Earth’s rotation. Therefore, the relative motion between the satellite and a user on the ground must be less than 4 km/s. Typical values around 1 km/s can be expected for the relative speed along the line of sight (range rate). The second consequence is the phenomena of “repeating ground tracks” every day. It is straightforward to calculate the time it takes for the satellite to complete one orbital revolution. The orbital period is approximately T = 11 hr 58 min. Therefore a GPS satellite completes 2 revolutions in 23 hr 56 min. This is intentional, since it equals the sidereal day, which is the time it takes for the Earth to rotate 360 o . (Note that the solar day of 24 hr is not 360 o , because during the day, the position of the Sun in the sky has changed by 1/365.25 of a day, or 4 min, due to the Earth’s orbit around the Sun). Therefore, every day (minus 4 minutes), the satellite appears over the same geographical location on the Earth’s surface. The “ground track” is the locus of points on the Earth’s surface that is traced out by a line connecting the satellite to the centre of the Earth. The