# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 6 Numerical Comparison  
The results obtained by the analytic formulas are now compared with a numerical code simulating the launcher trajectory with better accuracy. The equations of motion are the three-dimensional equations of flight in the relative velocity frame with state variables \((r,\lambda,L,V_{R},\gamma_{R},\psi_{R})\)  
\[\begin{array}{l}\dot{r}=V_{R}\sin\gamma_{R}\\ \dot{\lambda}=V_{R}\cos\gamma_{R}\frac{\cos\psi_{R}}{r\cos L}\\ \dot{L}=\frac{V_{R}\cos\gamma_{R}}{r}\sin\psi_{R}\\ \dot{V}_{R}=r\omega_{E}^{2}\cos L(\cos L\sin\gamma_{R}-\sin L\cos\gamma_{R} \sin\psi_{R})+\frac{f_{V}}{m}  
coordinates. After the vertical arc, to turn the trajectory from the vertical direction, a pitch manoeuvre starts with a fixed direction of thrust \(\alpha=\theta_{k}\) and ends when \(\gamma_{R}=\theta_{k}\). At this time (called pitch over), the gravity-turn trajectory starts. The pitch direction of thrust \(\theta_{k}\) is chosen so to have the prescribed value \(\gamma_{2_{f}}\) at the second-stage burn out. Then, the coasting arc duration is chosen to achieve the required orbit.  
Figures 6-8 show the analytical and numerical graphs of the velocity, flight-path angle and altitude during gravity turn. Note that at the end of the second stage the discrepancy between analytic and numeric results is rather small for the velocity, whereas a reduction of 15% occurs in the altitude due to the numerical integration of the drag force. In the numerical calculation the atmospheric density is represented by the USATMO 76 model and the drag coefficient is kept constant at an average value \(C_{D}=0.4\). The reference surface is \(S=\frac{\pi\,d}{4}\), with a diameter of \(d=2\) m. The drag effect reduces the payload mass that can be inserted into the reference orbit.  
In fact, the numerical results give insertion in the polar circular orbit of altitude 650 km of a payload of mass \(m_{pay}=390\) kg, vertical flight duration \(t_{\rm{p}}=5\) s, and pitch angle \(\theta_{k}=76\) deg (generating \(\gamma_{2_{f}}=26.5\) deg), and coasting time \(t_{c}=355\) s. The payload mass reduction with respect to the analytical results is about 4%. The evolution of the orbit parameters during the launch is shown in Figures 9 and 10, and Figure 11 shows the launcher ground track.  
Figure 6: The velocity plot: analytic and numeric results during the gravity-turn arc. Numerical velocity of the first and second stage is in black, analytic velocity of the first stage is in blue and second stage in orange line.  
Figure 7: The flight-path angle plot: analytic and numeric results during gravity-turn arc. Numerical flight-path angle of the first and second stage is in black, analytic flight-path angle of the first stage is in blue and second stage in orange line.  
Figure 8: The altitude plot: analytical and numerical results during gravity-turn arc. Numerical altitude of the first and second stage is in black, analytical altitude of the first stage is in the blue line and the second stage in the orange line.  
Figure 10: Out–of-plane orbit parameters during the ascent. First-stage parameters are in the blue line; coasting and second stage are in the orange line.  
Figure 9: Thein-plane orbit parameters during the ascent. First-stage parameters are the blue line; coasting and second stage are the orange line.  
Figure 12 shows the payload mass evaluation for polar circular orbits of different altitudes obtained by the analytic (blue curve) and numerical (orange curve) algorithms. Of course, the bigger differences are for orbits of lower altitude; however, the reduction in the payload mass does not exceed 10% of the value found analytically.