Fundamentals of Satellite Navigation - Understanding GPS: Principles and Applications - page 36

the spherical harmonic coefficients C lm and S lm through 360th degree and order. For
GPS orbit computations, however, coefficients are used only through degree and
order 12.
Additional forces acting on satellites include the so-called third-body gravita-
tion from the Sun and Moon. Modeling third-body gravitation requires knowledge
of the solar and lunar positions in the ECI coordinate system as a function of time.
Polynomial functions of time are generally used to provide the orbital elements of
the Sun and Moon as functions of time. A number of alternative sources and formu-
lations exist for such polynomials with respect to various coordinate systems (for
example, see [14]). Another force acting on satellites is solar radiation pressure,
which results from momentum transfer from solar photons to a satellite. Solar radia-
tion pressure is a function of the Sun's position, the projected area of the satellite in
the plane normal to the solar line of sight, and the mass and reflectivity of the satel-
lite. There are additional forces acting on a satellite, including outgassing (i.e., the
slow release of gases trapped in the structure of a satellite), the Earth's tidal varia-
tions, and orbital maneuvers. To model a satellite's orbit very accurately, all of these
perturbations to the Earth's gravitational field must be modeled. For the purposes of
this text, we will collect all of these perturbing accelerations in a term a d , so that the
equations of motion can be written as

d
dt

2

r

=∇ V + a (2.7)

d

2

There are various methods of representing the orbital parameters of a satellite.
One obvious representation is to define a satellite's position vector, r 0 = r ( t 0 ), and

v ( t 0 ), at some reference time, t 0 . Given these initial conditions,
we could solve the equations of motion (2.7) for the position vector r ( t ) and the
velocity vector v ( t ) at any other time t . Only the two-body equation of motion (2.4)
has an analytical solution, and even in that simplified case, the solution cannot be
accomplished entirely in closed form. The computation of orbital parameters from
the fully perturbed equations of motion (2.7) requires numerical integration.
Although many applications, including GPS, require the accuracy provided by
the fully perturbed equations of motion, orbital parameters are often defined in
terms of the solution to the two-body problem. It can be shown that there are six
constants of integration, or integrals , for the equation of two-body motion, (2.4).
Given six integrals of motion and an initial time, one can find the position and veloc-
ity vectors of a satellite on a two-body orbit at any point in time from the initial con-
ditions.
In the case of the fully perturbed equation of motion, (2.7), it is still possible to
characterize the orbit in terms of the six integrals of two-body motion, but those six
parameters will no longer be constant. A reference time is associated with two-body
orbital parameters used to characterize the orbit of a satellite moving under the
influence of perturbing forces. At the exact reference time, the reference orbital
parameters will describe the true position and velocity vectors of the satellite, but as
time progresses beyond (or before) the reference time, the true position and velocity
of the satellite will increasingly deviate from the position and velocity described by
the six two-body integrals or parameters. This is the approach taken in formulating

velocity vector, v 0 =