SOURCEFORGE » RALF » 2015 Building blocks for a LEO ground station

[[Introduction]]

[[State of Art of the Orbit Propagators]]

[[The Two-Lines Element]]

h2.  Distance elevation and azimuth calculation

h3.    ECI coordinates

The coordinates of the satellite given by the propagator are given in the Earth-Centered Inertial (ECI) coordinate system . This system is a cartesian coordinate system whose origin is located at the center of the earth (at the center of mass to be precise). The z axis is orthogonal to the equatorial plane pointing north, the x axis is pointing towards the vernal equinox, and the y axis is such that the system remains a direct orthogonal system The x and y axis are located in the equatorial plane as shown in the following figure.

!ECI.jpg!

This system is convenient to represent the positions and velocities of space objects rotating around the earth considering firstly that the origin of the system is the center of mass of the earth, and that the system does not rotate with the earth. Indeed, "inertial" means that the coordinate system is not accelerating, therefore not rotating: considering the way the three axis are defined, the system is fixed in space regarding the stars.

The problem here is that our ground is station is located on the surface of the earth, so its coordinates are given in the geodetic coordinate system, which is obviously a system that is rotating with the earth.

We had therefore two options: either calculate the coordinates of the ground station in the ECI coordinate system, or calculate the coordinates of the satellite in the geodetic coordinate system. We chose the first option .

h3. θ(τ)calculation

The main problem to go from the geodetic system to the ECI system is to calculate the angle between the observer meridian (longitude) and the vernal equinox direction. This angle, also called the local sidereal time depends on the star and not on the sun and that is why it is a bit touchy to calculate. For our implementation and tests, we relied on the algorithms and example given in the magazine _Satellite Time_ ,in the column Orbital Coordinate Systems, Part II. θ(τ) is actually define as the sum between the antenna east longitude and the greenwich sidereal time GST, which is the angle between the greenwich meridian and the vernal equinox that we will note  θg(τ).

θg(τ)can be calculated using the formula:

θg(τ) = θg(0h) + ωe·Δτ (1)

where Δτ is the number of seconds elapsed since 0h at the time of the calculation and ωe = 7.29211510 × 10-5 radians/second is the Earth's rotation rate. θg(0h) is the GST calculated at 0h that day and it is given as

θg(0h) = 24110.54841 + 864018.812866 Tu + 0.093104 Tu2 - 6.2 × 10-6 Tu3 (2)

where Tu = du/36525 and du is the number of days of Universal Time elapsed since the julian date 2451545.0 (2000 January 1, 12h UT1). Therefore, to calculate θg(τ), the first thing we need is the julian date of the day, which we can deduce from the julian date of the year as follow:

JD = Julian_Date_of_Year() + Number of day since the first of January;

This calculation is done in the VI julian date.vi where the calculation of the Julian date of the year is done using the Meeus' approach.

Once we had the Julian date, we could calculate du and therefore Tu, and that way we calculated θg(0h) using (2). From there, we were able to calculate θg(τ) using (1) and since we know the east longitude of the antenna we obtained θ(τ). Those calculations are done in the VI tetag.vi.

h3. Geodetic to ECI conversion

Once we had the local sidereal time, we were able to make the change of system using the following formulas:

x=Rcos θ

y=Rsin θ

z=Re sin φ

where Re = 6378.135 km, φ is the latitude of the antenna, θ is the local sidereal time and R = Re cos φ

This is done in the VI, Antenna coordinates ECI.vi.

h3. Distance

h3. Elevation and azimuth

h2. Link budget