## 2 Materials and Methods  
To detail the necessary aspects, we start from a particular case, based on a communication link between the ground and a GEO satellite. The aspects developed here will be transposed into multi-orbital contexts such as LEO-MEO, LEO-GEO, or MEO-GEO links. The efficiency of satellite communication depends crucially on the planning and optimization of the link budget. It is a detailed analysis of the signal losses and gains encountered from the transmission of the signal from the source to the destination, thus facilitating the evaluation of the system performance. Many quantities are involved in this process; a list of them can be seen in Figure 1.  
As it can be seen, the involved parameters can be divided into three categories: emission parameters (of the ground station), link parameters, and reception parameters (of the satellite). The Effective Isotropic Radiated Power (EIRP) is the fundamental emission parameter and represents the power that an isotropic antenna would need to emit to achieve the same power density that the actual antenna produces in a given direction to the satellite [12]. This parameter quantifies much information from the radio system contained in the ground station and depends on the power of the modulated signal at the output of the amplifier power \(P_{TxHPA}\), the amplifier backoff loss \(OBO_{TxHPA}\), the feeder loss \(L_{TxFeder}\), the radome loss \(L_{Radome}\), other existing losses \(L_{TxOther}\), and the gain of the transmitting antenna \(G_{Tx}\), as defined in Equation (1):  
\[EIRP[dBW]=P_{TxHPA}[dBW]-OBO_{TxHPA}[dB]-L_{TxFeder}[dB]-L_{TxOther}[dB]+G_{Tx}[ dBi]-L_{Radome}[dB] \tag{1}\]  
The link parameters are given by the link distance and the totality of the losses that may occur and are not related to the transmission-reception system but to the propagation medium. The distance between the satellite and the ground station is a determining factor in link budget calculations. It contributes the most to the size of the losses and is the key factor in determining the highest link loss, which is given by the free space path loss (\(FSPL\)). Figure 1: Graphic representation of the link budget. \(FSPL\) is the signal loss in free space without obstructions between the ground station and the satellite, and it is given by the formula:  
\[FSPL[dB]=20\log\biggl{(}\frac{4\pi fd}{c}\biggr{)} \tag{2}\]  
where \(d\) is the distance between the ground station antenna and satellite antenna, \(f\) is the signal frequency and \(c\) is the speed of the signal wave [13]. Other attenuations caused by propagation conditions that affect the link budget and performance of satellite communication systems are given by rain, gas, and fog/cloud attenuation. These attenuations can significantly reduce the quality and reliability of the signal transmitted over long distances. The calculation of rainfall attenuation follows the ITU rainfall model as outlined in [14]. This model determines the specific attenuation of the transmitted signal \(\gamma_{R}\), which is influenced by the rainfall rate, signal frequency, polarization, and the elevation angle of the signal path. The specific attenuation is determined with Equation (3):  
\[\gamma_{r}[dB/km]=kR^{\alpha} \tag{3}\]  
where \(R\) is the rain rate and \(k,\alpha\) are two coefficients that vary according to the signal's frequency, polarization, and the elevation angle. The total rain attenuation \(L_{r}\), of the signal path is given by:  
\[L_{r}[dB]=d_{eff}\gamma_{r} \tag{4}\]  
where \(d_{eff}\) is the effective distance over which the signal encounters attenuation due to rain and it depends on the actual distance \(d\) and on a long-term statistical rain rate, \(R_{0,01}\), which shows the rain rate that is exceeded 0.01% of the time. It is described in [15] as defined in Equation (5). \[d_{eff}=\frac{d}{1+\frac{d}{35e^{-0.005R_{0,01}}}} \tag{5}\]  
Atmospheric gas attenuation is a function of distance, frequency, temperature, atmospheric pressure, and water vapor density, defined in [16].