## 2 Materials and Methods
### Equations used
Duration of the satellite's flight over a given area:  
\[T=\frac{L}{v_{o}\cdot\cos\theta}, \tag{13}\]  
where L is the swath width, \(\mathrm{v_{o}}\) is orbital velocity, and \(\theta\) is the angle of inclination of the satellite's trajectory in reference to the equator [28]. The \(\cos\theta\) function adjusts the actual velocity of the satellite over Earth's surface along the parallel. **Determining the possible change in inclination**:  
Possible change in inclination:  
\[i_{m}=2\cdot\sin^{-1}(\frac{\Delta v}{2v_{0}}), \tag{14}\]where \(\Delta\)v is the increase in velocity, and v\({}_{\rm o}\) is the orbital velocity [7]. The formula results from the principles of orbital mechanics and geometry, which determine the change in the inclination of the orbit  
During a change in inclination, an impulse is introduced that changes the direction of orbital velocity. The change in inclination is proportional to the given impulse. As inclination is defined in reference to the equatorial plane, the value of arcsin enables to calculate the angle of change in trajectory depending on the applied impulse. Target inclination:  
\[i_{d}=i_{p}-i_{m}, \tag{15}\]  
where i\({}_{\rm p}\) is the initial inclination and i\({}_{\rm m}\) is the possible change in inclination. The formula results from the principle of adding or subtracting angles in the geometrical system of the orbit. The inclination change manoeuvre leads to shifting the orbital plane in reference to the equator by the i\({}_{\rm m}\) value, which is limited by the fuel and energy resources. The difference between the initial inclination and the possible change in inclination gives the target value, assuming that the change is subtracted from the initial inclination in the correct direction. **Determination of the weight of fuel necessary to perform the Hohmann transfer**  
Initial radius of the orbit:  
\[r_{p}=R_{Z}+h_{p}, \tag{16}\]  
where R\({}_{Z}\) is the radius of the Earth, and h\({}_{\rm p}\) is the initial altitude [7]. The formula results from the geometry of the orbit, where the initial radius of the orbit is the distance from the centre of the Earth to the satellite. Since the satellite is at the altitude h\({}_{\rm p}\) above the surface of the Earth, the total radius of the orbit is a sum of the radius of the Earth and this altitude. Target radius of the orbit:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]0  
where R\({}_{Z}\) is the radius of the Earth, and h\({}_{\rm d}\) is the target altitude [7]. The formula results from the geometry of the orbit. The radius of the target orbit is the distance from the centre of the Earth to the satellite. When the satellite reaches the target altitude, the radius of the orbit will be the sum of the radius of the Earth and this altitude. Velocity on the initial orbit:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]1  
where G is the gravitational constant 6.6743 \(\times\) 10\({}^{-11}\), M\({}_{Z}\) is the weight of the Earth 5.972 \(\times\) 10\({}^{24}\), and r\({}_{\rm p}\) is the radius of the initial orbit [7]. The formula results from the balance of forces on the orbit. On the orbit, the satellite moves in balance between the gravity force of the Earth and the centrifugal force that results from orbital movement. The formula is derived from [PERSON]'s second law of motion. After equalising these forces and simplifying, the formula for orbital velocity is obtained. Velocity on the target orbit:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]2  
where G is the gravitational constant 6.6743 \(\times\) 10\({}^{-11}\), M\({}_{Z}\) is the weight of the Earth 5.972 \(\times\) 10\({}^{24}\), and r\({}_{\rm d}\) is the radius of the target orbit [7]. Velocity on the transfer orbit at the perigee:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]3where G is the gravitational constant 6.6743 \(\times\) 10\({}^{-11}\), M\({}_{\rm Z}\) is the weight of the Earth 5.972 \(\times\) 10\({}^{24}\), r\({}_{\rm p}\) is the radius of the initial orbit, and a is the semi-major axis [7].