*3.2.3.2 ODCS SW*

### *3.2.3.2.1 Satellite orbit propagation (OP)*

To understand the idea of propagation of satellite orbit in Earth gravity field to the simplest, Keplerian motion propagator based on spherical Earth gravity field model might be used [9]; however more realistic results can be obtained with more accurate propagator, taking into account the second zonal harmonic *J*<sup>2</sup> in the function of approximation of Earth gravitational potential. The following equations of motion of satellite center of mass in Earth gravitational field can be considered [9]:

$$\begin{split} \ddot{\boldsymbol{x}} &= -\mu \frac{\boldsymbol{x}}{r^3} + A\_{f2} \left( \mathbf{15} \frac{\mathbf{x} \mathbf{z}^2}{r^7} - \mathbf{3} \frac{\mathbf{x}}{r^5} \right), \\ \ddot{\boldsymbol{y}} &= -\mu \frac{\boldsymbol{y}}{r^3} + A\_{f2} \left( \mathbf{15} \frac{\mathbf{y} \mathbf{z}^2}{r^7} - \mathbf{3} \frac{\mathbf{y}}{r^5} \right), \\ \ddot{\boldsymbol{z}} &= -\mu \frac{\boldsymbol{z}}{r^3} + A\_{f2} \left( \mathbf{15} \frac{\mathbf{z}^3}{r^7} - \mathbf{9} \frac{\mathbf{z}}{r^5} \right), \\ \boldsymbol{r} &= \sqrt{\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2}, \end{split} \tag{39}$$

where *x*, *y*, *z* are the Cartesian coordinates of satellite center of mass in inertial frame ECI, *<sup>r</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>p</sup> is the module of the radius vector from the center of Earth to satellite center of mass, *<sup>μ</sup>* <sup>¼</sup> <sup>3</sup>*:*<sup>986004418</sup> � 1014 *<sup>m</sup>*3*=<sup>s</sup>* <sup>2</sup> ½ � is the Earth gravitational constant, *AJ*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> *<sup>J</sup>*<sup>2</sup> � *<sup>R</sup>*<sup>2</sup> *<sup>e</sup>* is a constant, *J*<sup>2</sup> =0.00108263 is the second zonal harmonic coefficient in the raw of Earth potential function, and *Re* ¼ 6378137*:*00 *m* is the mean radius of the Earth at the equator.

These equations can propagate satellite position and velocity (*x*, *y*, *z* and *x*\_, *y*\_, *z*\_) in the inertial Cartesian ECI coordinate system if the initial parameters are initially set *x*<sup>0</sup> ¼ *x*ð Þ 0 , *y*<sup>0</sup> ¼ *y*ð Þ 0 , *z*<sup>0</sup> ¼ *z*ð Þ 0 and *x*\_ <sup>0</sup> ¼ *x*\_ð Þ 0 , *y*\_<sup>0</sup> ¼ *y*\_ð Þ 0 , *z*\_<sup>0</sup> ¼ *z*\_ð Þ 0 . They can be periodically determined from GPS or MCC TLM information. The propagation credibility time depends on orbit perturbations [9, 10, 27] and required accuracy. The most accurate and common ground propagators are NORAD Simplified Perturbation Model (SGP) propagators. NORAD SGP is used for proving to users twoline element (TLE) satellite orbital data. For the low Earth orbit (LEO), having altitude below 6000 km (period about 225 min), they provide position accuracy about 1 km within a few days that for many users is accurate enough and needs to be updated once or twice per week. Currently almost every satellite is equipped with GPS and its onboard propagators are practically continuously corrected with GPS (and sometimes MCC TLM) data that provide position within 10–100 m and velocity within 0.01–0.1 m/s accuracy range. Only some short periods of GPS data outage require orbit propagation. In addition to satellite position and velocity, OP calculates conventional orbital parameters (**Figure 14**) that can be computed with the following formulas [9]:

$$\begin{aligned} a &= \frac{\mu}{2\left[\frac{\mu}{r} - \frac{V^2}{2}\right]}\\ i &= \cos^{-1}\frac{h\_{xi}}{h} \\ \Omega &= \tan^{-1}\frac{h\_{xi}}{-h\_{yi}}\\ \nu &= \tan^{-1}\frac{\overline{r}\cdot\overline{V}}{p-r} \\ u &= \sin^{-1}\frac{z\_i}{r\sin i} \\ w &= u\cdot\theta \end{aligned} \tag{40}$$

where *h* ¼ *r* � *V* is the satellite orbital linear momentum vector, *a* is the satellite orbit semi-major axis, *h* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *h*2 *xi* <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> *yi* <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> *zi* <sup>q</sup> is the linear momentum module, *p* ¼ *<sup>a</sup>* <sup>1</sup>‐e2 ð Þ is the satellite orbit focal parameter, *<sup>i</sup>* is the satellite orbit inclination angle, <sup>Ω</sup> is the satellite orbit right ascension of ascending node angle (RAAN), *u*is the argument of latitude angle, *ν* is the satellite true anomaly angle, and *ω* is the satellite orbit argument of perigee angle (**Figure 27**).
