# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 2 First Sizing of a Launch Vehicle  
Achieving a single stage to orbit is still an infeasible task and space launchers have a number \(N\) of stages ranging from 2 to 4. Each stage has a total mass \(m_{i}\)  
\[m_{i}=m_{p_{i}}+m_{si}\]  
where \(m_{p_{i}}\) is the propellant contained within the tank of each stage and the structural mass coefficient of each stage is defined as  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]  
The total mass at lift off is  
\[M_{1}=\Sigma_{i=1}{}^{N}\,m_{i}+\left(m_{hs}+m_{pay}\right)\]  
with \(m_{hs}\) the mass of the heat shields (here released at the end of the gravity-turn phase) and \(m_{pay}\) is the satellite mass to be injected into orbit. At the time \(t_{b1}\) of the end of the first stage boost (burn time), the propellant mass \(m_{p_{1}}\) is consumed and the mass of the space rocket is  
\[M_{1f}=M_{1}-m_{p_{1}}\]  
The velocity reached at burn time \(t_{b1}\) can be computed integrating the acceleration  
\[\dot{V}=\frac{T}{m}\,\cos\alpha-\frac{\mu}{r^{2}}\,\sin\gamma-\frac{1}{2}\rho V ^{2}\frac{C_{D}\,S}{m} \tag{2}\]  
here \(T\) is the thrust intensity with a direction determined by the angle \(\alpha\) taken from the velocity \(\dot{V}\). The velocity direction is determined by the flight-path angle \(\gamma\) taken from the local horizon direction \(\dot{\theta}\), see Figure 1. The variable \(r\) is the radius distance from the centre of the Earth to the launcher centre of mass and \(\mu\) is the Earth gravitational constant. The atmospheric density is denoted by \(\rho\) and \(C_{D}\) is the launcher drag coefficient computed with respect to the reference surface \(\dot{S}\). To underline the thrust's contribution to the variation in velocity and consider all the other terms as \"losses\", let us write (2) as  
\[\dot{V}=\frac{T}{m}-\frac{\mu}{r^{2}}\,\sin\gamma-\frac{1}{2}\rho V^{2}\frac {C_{D}\,S}{m}-\frac{T}{m}\left(1-\cos\alpha\right)\]  
Integrating the above acceleration between time zero and \(t_{b1}\), one obtains the velocity after the first-stage boost  
\[\Delta V_{1}=\int_{t_{0}}^{t_{b1}}\frac{T}{m}\,dt-\int_{t_{0}}^{t_{b1}}\frac {\mu}{r^{2}}\,\sin\gamma\,dt-\int_{t_{0}}^{t_{b1}}\frac{1}{2}\rho V^{2}\frac {C_{D}\,S}{m}\,dt-\int_{t_{0}}^{t_{b1}}\frac{T}{m}\left(1-\cos\alpha\right)dt \tag{3}\]Equation (3) is called the \"equation of losses\"  
\[\Delta V_{1}=\Delta V_{7z}-\Delta V_{grav}-\Delta V_{drag}-\Delta V_{mis}\]  
where the first variation of velocity due to the engine boost is called the Tziolkowski variation of velocity and the other velocities subtracted to \(\Delta V_{7z}\) are called gravity, drag and misalignment losses, respectively. Average thrust is defined by \(T=gIsp\,\dot{m}\), where \(\dot{m}\) is equal to the mass rate \(\dot{m}=-\frac{dm}{dt}\). Inserting \(T=-gIsp\,\frac{dm}{dt}\) in (3), we obtain the Tziolkowski velocity variation provided by the first-stage engine  
\[\Delta V_{7z}=gIsp\,\log(\frac{M_{1}}{M_{1f}})\]  
Introducing the subrocket structural mass ratios  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]0  
the variation of the velocity provided by the engines of a multistage rocket is  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]1  
The values of the specific impulses \(Isp_{k}\) depend on the kind of propellant, for instance liquid or solid. Figure 2 shows the specific impulses for stages with solid propellant: these values are taken from the database [18] and we have average values  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]2  
Other parameters defined by technical constraints are the structural mass ratio of each stage (1): Figure 3 shows these values for a number of stages taken from the database [18], and we have average values  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]3  
\[e_{i}=\frac{m_{si}}{m_{i}} \tag{1}\]4 The state variable of the launch trajectory. The inertial frame (\(\dot{r}_{0},\dot{\theta}_{0}\)) has origin on the centre of the Earth (in dark blue). The orbit frame (\(\dot{r},\dot{\theta}\)) has origin on the launcher centre of mass.