## 2 Materials and Methods
### Equations used
The formula results from the principle of energy conservation on Kepler's orbit, where the sum of kinetic and potential energy is constant. The formula takes into account the gravitational relation between velocity and the radius of the initial orbit and the shape of the orbit (the semi-major axis). Velocity on the transfer orbit at the apogee:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]4  
where 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\({}_{\rm d}\) is the radius of the target orbit, and a is the semi-major axis [7]. First increase in velocity:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]5  
where v\({}_{\rm tp}\) is the velocity on the initial orbit at the perigee and v\({}_{\rm p}\) is the velocity on the initial orbit [7]. This formula describes the first velocity impulse required to transfer from the initial orbit to the transfer orbit. It results from the difference in the velocity that the satellite must achieve in the perigee to enter the transfer orbit. Second increase in velocity:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]6  
where v\({}_{\rm d}\) is the velocity on the target orbit and v\({}_{\rm ta}\) is the velocity in the apogee on the transfer orbit [7]. Total increase in velocity:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]7  
where \(\Delta v_{1}\) is the first increase in velocity and \(\Delta v_{2}\) is the second increase in velocity [7]. This formula defines the total increase in velocity required to perform the complete Hohmann transfer. The first impulse changes the velocity to move from the initial orbit to the transfer orbit, and the second impulse adjusts the velocity to transfer from the transfer orbit to the target orbit. The total increase in velocity is a sum of these two impulses that constitutes the total fuel cost of the manoeuvre. Required weight of fuel:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]8  
where m\({}_{\rm p}\) is the initial weight of the satellite with fuel, v\({}_{\rm e}\) is the effective exhaust velocity, and \(\Delta v_{\rm total}\) is the total increase in velocity [7]. This formula is used to calculate the weight of fuel required to perform an orbital manoeuvre based on [PERSON]'s rocket equation. It takes into account the effective exhaust velocity, the total increase in velocity, and the initial weight of the satellite. The formula describes the amount of fuel consumed to achieve the required change in velocity, taking into account the characteristics of the propulsion system and fuel efficiency. **Calculating the energy required to tilt the satellite**  
Energy consumed by reaction wheels:  
\[v_{e}=I_{sp}\cdot g, \tag{8}\]9where P is the power collected by reaction wheels and t is the duration of operation of these reaction wheels [28]. The formula allows for the estimation of the total energy consumption during operations related to changing the orientation of the satellite. Loading time:  
\[\Delta v=v_{e}\cdot\ln\frac{m_{p}}{m_{k}}, \tag{9}\]0  
where E is the energy consumed by reaction wheels and P\({}_{\text{t}}\) is the charging power of reaction wheels [28]. The formula enables to estimate how long the satellite must use its solar panels or other power sources to restore the full operational capacity. _Presentation of the constant parameters for the analysed manoeures_  
The area that was analysed for the purposes of this research project has a surface of approx. 385 km2, which corresponds to an average big city in Poland or the surface area of a typical national park. These types of areas often require various analyses, such as analysing the degree of urbanisation or the quality of trees and waters. For the purposes of this study, it was assumed that the analysed area would be the Kampinos National Park, situated near Warsaw, at the latitude of 52.2297\({}^{\circ}\) = 0.911 rad. The swath width observed by the sensor mounted on the satellite depends directly on the optical parameters of the sensor and on the altitude of the orbit, and indirectly on the weight of the satellite.