Bernese GPS Software Version 5.0 - Bernese GNSS Software

2. Fundamentals
2.2.2.1 The Keplerian Orbit
The mathematical description of a satellite orbit would be very simple if the gravity field of
the Earth were spherically symmetric, if the Earth were the only celestial body acting on
the satellite, and if, moreover, non-gravitational forces like air-drag and radiation pressure
would not exist. Maybe life on Earth would be problematic in this case, however.
Under these circumstances the geocentric orbit r(t) of a satellite in inertial space is described
by a simple differential equation system of second order in time, the so-called equations of
motion for the case of the two-body problem (actually even a reduced version of the twobody
problem because we will always be allowed to neglect the satellite’s mass for the
gravitational attractions):
¨r = −GM r
, (2.7)
3
where GM is the product of the constant of gravity and the mass of the Earth, r is the
length of the geocentric radius vector r of the satellite.
It is well known that the solution of the equations of motion (2.7) is either an ellipse, a
parabola, or a hyperbola. We are obviously only interested in the first type of solutions. In
Figure 2.8 we see one possible set of six parameters describing the orbit. Exactly this set is
used for orbit characterization in the Bernese GPS Software. Let us make a few comments
concerning these orbital elements:
a is the semimajor axis of the orbit, defining the size of the orbit.
e is the numerical eccentricity or simply eccentricity of the orbit, describing the shape of
the orbit, i.e., the deviation from circularity.
i is the inclination of the orbital plane with respect to the equatorial plane.
Ω is the right ascension of the ascending node, i.e., the angle between the direction to the
vernal equinox (X-direction in Figure 2.8) and the intersection line of the satellite’s
orbital plane with the equatorial plane (in the direction of the satellite crossing the
equatorial plane from the southern to the northern hemisphere). i and Ω are the Eulerian
angles defining the orientation of the orbital plane in the equatorial system.
b
a e
E
a
r
v
Satellite
Earth Perigee
r
X
Earth
Figure 2.8: The set of orbital elements a,e,i,Ω,ω,u0.
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Ω
Z
u o
ω
v o
i
Perigee
Satellite
Y