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Fundamentals
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

The user receiver computes the satellites coordinates from the information broadcast by the GNSS satellites in the navigation messages.

Two different approaches are followed by GPS/Galileo and GLONASS satellites to account for satellite orbit perturbations. Those approaches define their messages contain.

In the case of GPS or Galileo satellites, the orbits are seen as Keplerian in first approximation, and the perturbations are treated as temporal variations in the orbital elements.

Indeed, an extended set of sixteen quasi-Kleperian parameters (see table (1) in GPS and Galileo Satellite Coordinates Computation) is broadcast to the user in the navigation message and regularly updated. This expanded set consists of the six orbital elements $\displaystyle(a(t),e(t),i(t),$ $\displaystyle \Omega (t),\omega (t), M(t))$ plus three rate parameters to account for the linear changes with time $(\stackrel{\bullet}{\Omega},\stackrel{\bullet}{i},\Delta n)$, three pairs of sinusoidal corrections $\displaystyle(C_c,C_s)$ (i.e., $\displaystyle C_c\cos(2\phi)$, $\displaystyle C_s\sin(2\phi)$), and the reference ephemeris epoch $t_e$ (see article GPS and Galileo Satellite Coordinates Computation).

For GLONASS satellites, the navigation message broadcasts initial conditions of position and velocity $(\mathbb{\mathbf r}_0,\mathbb{\mathbf v}_0)$ and moon and solar gravitational acceleration perturbation vector components (see table (1) in GLONASS Satellite Coordinates Computation) to perform a numerical integration of the orbit. The integration is based on applying a 4$^{th}$-order Runge-Kutta method to the equation:

$\mathbb{\mathbf {\ddot r}}=\nabla V+\mathbb{\mathbf k}_{sun\_moon} \qquad \mbox{(1)}$

where $V$ is the potential defined by
$\begin{array}{ll} V= & \displaystyle \frac{\mu}{r}\left[ 1- \displaystyle \sum_{n=2}^{\infty}{\left(\frac{a_e}{r}\right)^n J_n\; P_n(\sin \phi)} \right .\\ & + \left. \displaystyle \sum_{n=2}^{\infty}{\displaystyle \sum_{m=1}^{\infty}{\left(\frac{a_e}{r}\right)^n \left[ C_{nm} \cos m\lambda + S_{nm} \sin m\lambda \right ] P_{nm}(\sin \phi)}}\right ] \end{array} \qquad \mbox{(2)}$

presented in Perturbed Motion and ($\mathbb{\mathbf k}_{sun\_moon}$) are the moon-solar accelerations expressed in an inertial coordinate system (see article GLONASS Satellite Coordinates Computation).

Note: In the differential equations system from GLONASS Satellite Coordinates Computation:

$\left\{ \begin{array}{l} \frac{dx_a}{dt}=v_{x_a}(t)\\ \frac{dy_a}{dt}=v_{y_a}(t)\\ \frac{dz_a}{dt}=v_{z_a}(t)\\ \frac{dv_{x_a}}{dt}=-\bar{\mu} \bar{x}_a +\frac{3}{2}C_{20}\bar{\mu} \bar{x}_a \rho^2(1-5 \bar{z}_a^2)+ Jx_am+Jx_as\\ \frac{dv_{y_a}}{dt}=-\bar{\mu} \bar{y}_a +\frac{3}{2}C_{20}\bar{\mu} \bar{y}_a \rho^2(1-5 \bar{z}_a^2)+ Jy_am+Jy_as\\ \frac{dv_{z_a}}{dt}=-\bar{\mu} \bar{z}_a +\frac{3}{2}C_{20}\bar{\mu} \bar{z}_a \rho^2(3-5 \bar{z}_a^2)+ Jz_am+Jz_as\\ \end{array} \qquad \mbox{(3)} \right .$

the term $C_{20}=-J_2=+\sqrt{5}\bar{C}_{20}$ is used instead of $J_2$ to keep the same expressions as in GLONASS-ICD.

Comment: At any epoch the state of motion of the satellite is given by six parameters: The position and velocity vector components $\displaystyle (\mathbb{\mathbf r},\mathbb{\mathbf v})$, or the six Keplerian elements $\displaystyle (a,e,i,\Omega, \omega,V)$; therefore, a point-to-point transformation can be done between them. The orbit elements are the natural representation of the orbit, because (in absence of perturbations) the motion along the orbit is described by a single parameter $\displaystyle(V(t))$. In presence of perturbing forces, time-varying Keplerian elements defining an ellipse tangent to the orbit at any epoch can be considered, i.e, an osculating orbit [footnotes 1].

## Notes

1. ^ From the Latin verb osculor (to kiss).