# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 3 A Second Step (Gravity Losses)  
At lift off, the launcher performs a vertical trajectory followed by a pitch manoeuvre to select the required orbit plane. Then, to limit the aerodynamic loads on the structure, any launcher is forced to follow a zero angle of attack trajectory (\(\alpha=0\)) during the atmospheric flight. This trajectory is called the gravity-turn trajectory, and it is performed by the first stages of space launchers. The trajectory is basically planar and the state variables can be the relative velocity, flight-path angle, altitude and range \((V,\gamma,h,s)\). The equations of motion are  
\[\left\{ \begin{array}{l}\dot{V}=\frac{T}{m}-\frac{h}{\
u^{2}}\sin\gamma-\frac {1}{2}\rho V^{2}\frac{C_{\rm D}S}{m}\\ \dot{\gamma}=\frac{s}{R_{E}}-\frac{\mu}{\
u^{2}}\cos\gamma+\frac{1}{2}\rho V \frac{C_{\rm L}S}{m}\\ \dot{h}=V\sin\gamma\\ \dot{s}=\frac{R_{E}}{I}\,V\cos\gamma\end{array} \right. \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Initial Mass \(M_{1}\)** & **Payload Mass \(m_{pay}\)** \\ \hline
18,000 kg & 100 kg \\ \hline
36,000 kg & 200 kg \\ \hline
54,000 kg & 300 kg \\ \hline
72,000 kg & 400 kg \\ \hline
90,000 kg & 500 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Minimumlift-off mass to inject the payload mass into circular polar orbit of altitude \(h=650\,\)km (average parameters (5) and (6), and launch site L = 30 deg). Figure 4: Therelative (dotted) and required (solid line) velocities to achieve circular orbits of altitude h = 800 km (black), 700 km (green), 600 km (blue), 500 km (red) and different inclinations  
In fact, these equations are singular under the lift-off condition; \(V=0,\gamma_{0}=\frac{p^{i}}{2}\). As observed in [14], this singularity generates an infinite number of solutions corresponding to the same lift-off condition: one can use this singularity to select among the different solutions the one with a prescribed final value, for instance a fixed flight-path angle \(\gamma_{f}\)[15]. An analytic formula for the velocity \(V\), due to [PERSON] and [PERSON], was obtained under the following hypotheses:  
* Constant gravity;
* Negligible drag and lift effect;
* Constant thrust-over-mass ratio \(n=\frac{T}{g\,m}\). The last of these is the strongest assumption, and is handled considering an average value \(\overline{n}\) of the thrust-to-weight ratio. This average value can be computed using the formulas for the variation of mass \(m=m_{0}-m_{d}\,t=m_{0}(1-q\,t)\) and the definition of the mass ratio \(U=\frac{m_{0}}{m_{0}}\), which verifies \(U=1-q\,t_{b}\). Then  
\[\bar{n} = \frac{1}{t_{b}}\int_{0}^{tb}\frac{T}{mg}(t)\,dt=\frac{T}{g\,t_{b} \,m_{0}}\int_{0}^{tb}\frac{dt}{1-qt}=\] \[= \frac{n_{0}}{t_{b}}\frac{1}{-q}\left[\ln(1-qt)\right]_{0}^{t_{b}} =\frac{n_{0}}{-q\,t_{b}}\ln(1-qt_{b})=\] \[= -\frac{n_{0}}{1-U}\ln(U)=\frac{n_{0}}{1-U}\ln(\frac{1}{U})\]  
where \(n_{0}=\frac{T}{g\,m_{0}}\) is the initial thrust-to-weight ratio. Now two transformations are defined:  
1. From the flight-path angle variable \(\gamma\) to the kick angle \(\chi=\frac{\pi}{2}-\gamma\). 2. From the kick angle \(\chi\) to the variable \(z=\tan\frac{\chi}{2}\). By the transformation in Equation (1) we have the following differential equations for the velocity and kick angle  
\[\left\{ \begin{array}{l}\bar{V}=-g\,\cos\chi+g\overline{n}\\ \dot{\chi}=\frac{g}{\bar{V}}\sin\chi\end{array} \right. \tag{8}\]  
Now the following trigonometrical equations are recalled  
\[\cos\chi = \frac{1-\tan^{2}\frac{\chi}{2}}{1+\tan^{2}\frac{\chi}{2}}=\frac{ 1-z^{2}}{1+z^{2}}\] \[\sin\chi = \sin\Bigl{(}2\,\frac{\chi}{2}\Bigr{)}=2\sin\frac{\chi}{2}\cos \frac{\chi}{2}=2\tan\frac{\chi}{2}\cos^{2}\frac{\chi}{2}=\frac{2z}{1+z^{2}}\]  
showing that the transformation to the \(z\)-variable allows us to transform the trigonometric function in rational functions. Then, Equation (8) becomes  
\[\left\{ \begin{array}{l}\dot{V}=-g\,\hbar-g\,\frac{1-z^{2}}{1+z^{2}}\\ \dot{z}=\frac{g}{V}\,z\end{array} \right.