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Geodetic constants Algorithm of orbit computation Keplerian elements at reference time Change rates, correction values and correction terms Computation of the orbit points Orbit computation from a GPS almanach Satellite orbit in the starfixed system Satellite orbit velocitiy In the library

From ephemeris or almanac data of GNSS satellites (e.g. GPS) discrete orbit points are computed at a specified time grid. The computation is based on the GPS Interface Specification IS-GPS-200 and the GALILEO Signal in Space Interface Control Document.

≡ START First steps Deutsch Geodetic constants

As usual in geodesy, the geocentric gravitational constant GM refers to the total mass of the Earth, including the atmosphere.

At the time of introduction of GPS, the geocentric gravitational constant GM was not as exactly known as it is today. At this time the value of GM = 3986005·10⁸ m³/s² was used, while today it is mostly replaced by the more accurate value of GM = 3986004.418·10⁸ m³/s² , which is defined in the . This new value is also used for GALILEO. To allow older GPS receivers to compute the orbits correctly nonetheless, the broadcast ephemeris data of GPS satellites are still adapted to the old value.

The angular velocity of Earth′s rotation must refer to the fixed stars. In the the value of ω_E = 7.2921151467e-5 rad/s is defined and should routinely be used.

≡ START First steps Deutsch Algorithm of orbit computation

The algorithm is described in the official document GPS Interface Specification IS-GPS-200 in table 30 and in the official GALILEO Signal in Space Interface Control Document in table 58. The symbols used there differ slightly from ours.

≡ START First steps Deutsch Keplerian elements at reference time

By the following six orbital elements an unperturbed Keplerian orbit is defined, i.e. a spatial ellipse fixed with respect to the stars and the current position (known as ''anomaly'') within this ellipse.

a	semi-major axis of orbital ellipse
e	numerical eccentricity of orbital ellipse
i	orbital inclination angle
ω	argument of perigee
Ω	ascension of ascending node
M	mean anomaly

There are different options for specification of the size of the ellipse and the anomaly. The official GPS definition uses die square root of the semi-mayor axis √a and the mean anomaly M Input options of alternative orbit parameters are offered as well:

instead of semi-major axis of orbital ellipse	instead of mean anomaly
T0 revolution period	E0 eccentric anomaly
n0 mean motion	v0 true anomaly

The Keplerian elements refer to the time instance t_oe, which is given in seconds after the beginning of the GNSS week, but not the right ascension of the ascending node Ω_o which refers to the beginning of the GNSS week t=0 .

≡ START First steps Deutsch Change rates, correction values and correction terms

If an unperturbed Keplerian orbit is desired, only these six Keplerian are needed. Change rates, correction values and correction terms of these elements are needed to model satellite orbit perturbations.

For mean precisions, e.g. for the computation of satellite`s rise and set times, at least the rate of right ascension dΩ/dt is required, except if the orbit should be computed at the beginning of the week. For GPS the reference value -2.6 semi-circles/s = -8.168e-9 rad/s is used, which specifies the approximate nodal precession caused by the Earth´s flattening. For GALILEO the corresponding value is -0.02764398°/d = -5.584e-9 rad/s.

For high precisions, e.g. for the computation of the receiver position from observations, also the other change rates, correction values and correction terms are required. They are provided as broadcast ephemeris data in the navigation message or as precise ephemeris data by an orbital service.

≡ START First steps Deutsch Computation of the orbit points

We compute the positions of the satellite reference point in the earth fixed righthanded cartesian coordinate system at the desired GNSS system time instances. Those time instances are defined by three parameters:

• start time of orbit computation t₁ (=1st orbit point) in seconds referring to the beginning of the current GNSS week
• time increment of orbit computation Δt in seconds
• number of orbit points to be computed

If only one should be computed, the last two parameters can be missing.

The beginning of a GNSS week t=0 is always sunday 0:00:00 in the GNSS system time. Note that this time differs from the universal time coordinated (UTC) by leap seconds.

If j is the counter of the points, the time instances with respect to t_oe as origin of the time axis are generated according to the following scheme:

δt_j = t₁ + (j-1)·Δt − t_oe

The following column formats determine the appearance of the :

• pointname coordinates: The times since beginning of the week are used as point names.
• pointname code coordinates: The points are automatically named. The times since beginning of the week are used as point codes.
• coordinates: The points are automatically named. The times are not displayed.

≡ START First steps Deutsch Orbit computation from a GPS almanach

Given is the following GPS almanach in YUMA format:

******** Week  297 almanac for PRN-02 ********
ID:                         02
Health:                     000
Eccentricity:               0.9529113770E-002
Time of Applicability(s):   589824.0000
Orbital Inclination(rad):   0.9551331376
Rate of Right Ascen(r/s):  -0.8183198006E-008
SQRT(A)  (m 1/2):           5153.635742
Right Ascen at Week(rad):   0.1038484770E+001
Argument of Perigee(rad):   1.827911506
Mean Anom(rad):             0.2496773193E+001
Af0(s):                    -0.2574920654E-004
Af1(s/s):                   0.0000000000E+000
week:                       297

We desire the positions of satellite 02 at the full hours of the last three days of the GPS week 297.

and Compute

≡ START First steps Deutsch Satellite orbit in the starfixed system

usually computes discrete orbit points in the earth fixed (rotating) righthanded cartesian coordinate system ECEF. However, if you want to obtain the orbit points in the star fixed (quasi inertial) system ECSF, simply specify for the angular velocity of Earth´s rotation ω_E = 0 . Then you obtain a system with axes coincident with the earth fixed system at the beginning of the week and then kept star fixed.

≡ START First steps Deutsch Satellite orbit velocitiy

are computed in the form of of discrete orbit points on a time grid. If you want to obtain the orbit velocity, transfer the coordinate list to and compute it as a polyline. Then you obtain the spatial distances of consecutive orbit points as side lengths of the polygon. If you chose e.g. 1 second as a computation time increment Δt then the side lengths are immediately velocities in metres/second.

Usually you obtain the velocities in the earth-fixed (rotating) coordinate system. If you want to obtain them in the starfixed system, please apply the previous trick. If you do this for the Orbit computation from a GPS almanach (time increment 1h), then you obtain orbit velocities between 13600 km/h and 14000 km/h.

Did you know? and also compute a Circle through three points.

≡ START First steps Deutsch