## 2 Materials and Methods
### Impulsive Manoeuvres
#### 2.2.2 Manoeuvre of Changing the Orbital Plane  
Changing the orbital plane is often necessary in situations when the launching point does not allow the spacecraft to reach the desired orbit directly. The latitude of the launching point has a significant influence on the inclination of the orbit: a satellite launched from the given location will automatically move on an orbit whose inclination is equal or higher than the latitude of the location. Due to that, in order to reach an orbit with an inclination other than that which results from the position of the starting point, it may become necessary to conduct a manoeuvre to change the plane of the orbit. These manoeuvres require a large \(\Delta\)v, which translates into high fuel consumption [7]. Changing the orbital plane consists of modifying the direction of the perpendicular component of the velocity of the spacecraft. The angle between the initial and target perpendicular component of velocity, denoted as \(\delta\), is a dihedral angle and it is presented in Figure 3.  
Similarly, as for radial velocity, the perpendicular component of velocity may also change its value during the inclination-changing manoeuvre. Orbits that have a common centre F do not have to be coplanar and are usually on different planes. Figure 3 shows two  
Figure 3: Manoeuvre changing the orbital plane—diagram. Created based on [7]. (**a**) Two non-coplanar orbits around F. (1) Initial orbit; (2) Final orbit. (**b**) View along the line of intersection of two orbital planes. (1) Plane of the initial orbit; (2) Plane of the final orbit. such orbits and their intersection line, marked as BD. Points A and P represent, respectively, the apoapsis and periapsis of the orbits. As the common centre F is on both orbital planes, it must also be located on their intersection line. In order for the spacecraft being on the initial orbit (orbit 1) to be able to change the plane to the target orbit (orbit 2) with a single \(\Delta\
u\) manoeuvre, it is necessary to conduct the manoeuvre at a point where the initial orbit intersects the target orbit. Such opportunity exists only in points B and D, which are situated on the intersection line of two orbital planes [7]. Required change in velocity  
General formula to calculate \(\Delta\
u\) for a change in the orbit plane [7]:  
\[\Delta v=\sqrt{\left(v_{r_{2}}-v_{r_{1}}\right)^{2}+v_{\perp 1}{}^{2}+v_{\perp 2}{}^{2} -2v_{\perp 1}v_{\perp 2}{\rm cos}\,\delta} \tag{4}\]  
To minimise \(\Delta\
u\) for the given dihedral angle \(\delta\), two conditions have to be met:  
The radial velocity component should remain constant, i.e., \(v_{r_{2}}-v_{r_{1}}=0\);  
The perpendicular velocity component should be as small as possible. Optimum conditions occur when the initial and target orbits intersect at the apoapsis. As shown in Figure 4, if the manoeuvre is conducted in the apoapsis, the radial velocity component is \(v_{r_{2}}=v_{r_{1}}=0\) for both orbits. Additionally, in the apocentre, the azimuthal velocity components are the lowest, which allows to minimise the value of \(\Delta\
u\)[7]. \[\Delta v=\sqrt{v_{1}{}^{2}+v_{2}{}^{2}-2v_{1}v_{2}{\rm cos}\,\delta}, \tag{5}\]  
If there is no change in velocity, i.e., \(v_{2}=v_{1}\), the equation takes the following form:  
\[\Delta v_{\delta}=2v{\rm sin}\,\frac{\delta}{2}, \tag{6}\]  
where the lower index of \(\delta\) reminds us that this is the \(\Delta\
u\) for the exclusive rotation of the velocity vector by angle \(\delta\)[7]. In conclusion, the manoeuvre of changing the orbit plane, also known as changing the inclination, is one of the most energy consuming orbital manoeuvres. Its aim is to change the inclination of the orbit, i.e., the angle of inclination of the orbital plane in relation to the reference plane, usually the plane of the planet's equator (for satellites orbiting around the Earth) or the plane of the ecliptic (for planets and objects in the solar system).