## 4 Propulsion System Design and Selection
### Propulsion System Selection
#### 4.1.2 Pulsed Plasma Thruster Calculations  
The European space agency has published the minimum requirements a CubeSat propulsion system should satisfy, this is shown by Table 25.  
A vast majority of launched CubeSat integrate passive propulsion systems and do typically not incorporate thruster technology [2]. Usually Magnetorquer and momentum wheel devices are implemented; however, these devices alone are not able to change the altitude of the CubeSat's orbit. They are only able to maintain the release orbit from the P-POD deployer [3]. Space debris is becoming an international issue of concern, and legislation is currently under development requiring satellites to de-orbit into a \"graveyard\" cycle, reducing the density of space debris at commonly used operating altitudes [61].  
Atmospheric drag acting upon the CubeSat will lead to the decay of the satellites orbit.  
\[F_{drag}=0.5*\rho*V^{2}*C_{D}*A_{sat} \tag{2}\]  
The \(C_{D}\) value for most satellites is assumed to be 2.2 [4]. The velocity of the CubeSat's natural orbit at 600 km is found using:  
\[v=\sqrt{\frac{G*M}{r}}=\sqrt{\frac{6.67408*10^{-11}*5.9723*10^{24}}{6.378137*10^ {6}+600*10^{3}}}=7557\text{ ms}^{-1} \tag{3}\]  
Therefore, the CubeSat is travelling at approximately 7.56 km/s.  
Assuming the satellite orbit \"sideways\" with the 100 mm \(\times\) 100 mm face cutting through the incoming air the \(A_{sat}=0.1\) m\({}^{2}\). The density of air at 600 km is taken to be \(1.03*10^{-14}\) kgm\({}^{-3}\)[6].  
Therefore, the force due to drag is equal to:  
\[F_{drag}=0.5*1.03*10^{-14}\text{ kgm}^{-3}*7557\text{ ms}^{-1}*2.2*0.1=8.562081*10^{-12}\text{ N}\]  
Assuming the CubeSat is orientation \"upwards\" the \(A_{sat}=0.3\) m\({}^{2}\). This gives a revised value of drag of \(2.5686243*10^{-11}\) N.  
The required graveyard orbit altitude the propulsion system must ensure the CubeSat reaches is found using:  
\[r_{orbit}=\frac{r_{o}}{\left((1+\frac{F_{drag}}{M_{sat}})*\sqrt{\frac{r_{o}}{ r_{auth}}}*t)\right)2} \tag{4}\]  
\[r_{orbit}=\frac{6,978,137}{\left(1+\frac{8.562081*10^{-12}}{4}*\sqrt{\frac{6.97 8,137}{5.9723*10^{24}*6.67408*10^{-11}}*5790}\right)^{2}}=513,000\text{ m}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Description** & **ESA CubeSat Requirement** \\ \hline Micro thruster module dry mass budget (kg) & 0.150–1.0 \\ Number of micro thruster per CubeSat & 1–6 \\ Power required by full assembly (W) & 1–10 \\ Buss Voltage (V) & 5–8 \\ Volume required by full assembly (m\({}^{3}\)) & 0.0001–0.0125 \\ Thrust Range in (micro N) & 1–1000 \\ Minimum impulse bit (Micro Ns) & 5 \\ Specific impulse (s) & 60–1000 \\ \(\Delta V\) (m s\({}^{-1}\)) & 1–60 \\ \hline \hline \end{tabular}
\end{table}
Table 25: ESA CubeSat Propulsion System Requirements [4].  
The value of \(\Delta V\) required to maintain the CubeSat's altitude is given by:  
\[\Delta V_{orbit}=\pi*\frac{C_{D}*A_{sat}}{M_{sat}}*\rho*\tau_{orbit}*V_{sat} \tag{5}\]  
\[\Delta V_{orbit}=\pi*\frac{2.2*0.1}{4}*1.03*10^{-14}*513*10^{-3}*7557=6.995*10^{- 6}\ \text{ms}^{-1}\]  
The total amount of propellant required for the mission duration depends on the orbital maneuvers required to be performed. This is individual to each mission. The total mass of propellant required is equal to:  
\[M_{ propellant}=\frac{L}{g_{o}*I_{sp}*S} \tag{6}\]  
\[Where:\ L=I_{x,y,z}*\omega \tag{7}\]  
The number of pulses the thruster can therefore deliver is given by the total mass divided by the mass of a singular pulse:  
\[N_{pulses}=\frac{M_{propellant}}{M_{Bit}} \tag{8}\]  
The energy stored in the thruster's capacitor is equal to:  
\[E=0.5*C*V^{2} \tag{9}\]