## 2 Materials and Methods
### Equations used  
Acceleration of gravity at the altitude of h\({}_{\text{P}}\):  
\[g=\frac{G\cdot M_{\text{Z}}}{\big{(}R_{\text{Z}}+h_{\text{P}}\big{)}^{2}}, \tag{7}\]  
where G is the gravitational constant 6.6743 \(\times\) 10\({}^{-11}\), M\({}_{\text{Z}}\) is the weight of the Earth 5.972 \(\times\) 10\({}^{24}\), R\({}_{\text{Z}}\) is the radius of the Earth 6371, and h\({}_{\text{P}}\) is the initial altitude. The equation is derived directly from the [PERSON]'s law of universal gravitation. The value of g decreases with the increase in h\({}_{\text{P}}\), as the force of gravity decreases proportionally to the square distance from the centre of the mass of the Earth [25]. Effective exhaust velocity:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]  
where I\({}_{\text{sp}}\) is the specific impulse, and g is the acceleration of gravity. This formula describes the efficiency of a rocket engine [26]. Increase in velocity:  
\[\Delta v=v_{e}\cdot\ln\frac{m_{p}}{m_{k}}, \tag{9}\]  
where v\({}_{\text{e}}\) is the effective exhaust velocity, m\({}_{\text{P}}\) is the initial weight of the satellite with fuel, and m\({}_{\text{k}}\) is the final weight of the satellite without fuel [26]. This formula results from the Tsiolkovsky equation that describes the relation between the increase in the velocity of a rocket, its effective exhaust velocity, and the ratio of the initial and final weight. The weight of the fuel consumed to conduct the manoeuvre:  
\[m_{f_{spont}}=m_{p}-m_{k}=m_{p}\Big{(}1-e^{-\frac{\alpha\mathrm{E}}{\mathrm{v}} }\Big{)}, \tag{10}\]  
\(\mathrm{m_{p}}\) is the initial weight of the satellite with fuel, \(\mathrm{m_{k}}\) is the final weight of the satellite without fuel, \(\mathrm{\Delta v}\) is the increase in velocity, and \(\mathrm{v_{e}}\) is the effective exhaust velocity [26]. This formula was obtained by transforming the Tsiolkovsky equation and it describes the mass of fuel consumed during an orbital manoeuvre. The above formula results from the logarithmic relationship between the increase in velocity and the ratio of the weights before and after burning the fuel. The formula takes into account the fact that the amount of consumed fuel increases with the required increase in velocity and decreases for higher values of effective exhaust velocity, which reflects the improved efficiency of the propulsion system. Due to that, the formula enables a precise calculation of the demand for fuel depending on the parameters of the manoeuvre and of the propulsion system. Duration of the impulse:  
\[t=\frac{m_{f_{spont}}\cdot v_{e}}{F}, \tag{11}\]  
where \(\mathrm{m_{f_{spont}}}\) is the weight of consumed fuel, \(\mathrm{v_{e}}\) is the effective exhaust velocity, and F is the thrust force [26]. This formula describes the duration of the impulse that is required to burn a specific amount of fuel during an orbital manoeuvre. The formula is derived from [PERSON]'s second law of motion and the definition of the thrust force. Thrust force is related to the amount of fuel burnt in a unit of time and the effective exhaust velocity. Therefore, time may be determined as the ratio of the total weight of fuel to the rate at which it is burnt. In this way, the formula demonstrates that the duration of the impulse depends both on the efficiency of the propulsion system and on the thrust power available. **Duration of the satellite's flight over a given area**:  
Inclination of the satellite's trajectory in reference to the equator:  
\[\theta=arcsin\frac{\cos\varphi}{\sin i}, \tag{12}\]  
where \(\varphi\) is the latitude, and i is the inclination of the orbit [27]. The arcsin function is used, because the \(\theta\) angle is based on the trigonometric proportion \(\sin\theta=\frac{\cos\varphi}{\sin i}\) between latitude and inclination. The application of this function enables us to determine the actual angle of inclination of the satellite's trajectory in reference to the parallel.