# A Brazilian Space Launch System for the Small Satellite Market
## 2 Methodology and Launch Vehicle Configuration
### Vehicle Performance
Rocket total velocity and altitude are given by  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]3  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]4  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]5  
Aerodynamic coefficients dependency on the velocity (Mach number) are given by the following relations [13; 14]:  
\[c_{x}=\left\{ \begin{array}{ll}0.290&0\leq M\leq 0.8\\ M-0.51&0.8\leq M\leq 1.068\\ 0.091+0.5M^{-1}&M\textgreater 1.0680\end{array} \right. \tag{16}\]  
\[c_{y}=\left\{ \begin{array}{ll}2.8&0\leq M\leq 0.25\\ 2.8+0.477(M-0.25)&0.25\leq M\leq 1.1\\ 3.18-0.66\,(M-1.1)&1.1\leq M\leq 1.6\\ 2.85+0.35\,(M-1.6)&1.6\leq M\leq 3.6\\ 3.55&M\textgreater 3.6\end{array} \right. \tag{17}\]  
The system of equations of the rocket flight trajectory is solved with the assumptions of: (i) gravitational acceleration on the active flight regime is constant; (ii) angles of attack are small, thus, \(\sin\alpha\approx\alpha\) and \(\cos\alpha\approx 1\); (iii) rocket flight on the active trajectory is defined by the function \(\vartheta(t)\). The gravitational model of the Earth and the characteristics of the standard atmosphere are also employed over the calculations. In addition, for each specific launch vehicle, initial input data includes the number of stages, the distribution of the initial dry mass and propellants over the stages, the law of change in propellant consumption for the main and attitude control engines, as well as some other characteristics (specific impulse of engines, aerodynamic coefficients, change of the center of mass during the flight, center of pressure, points of application of control forces, control system model and its coefficients). The launch trajectory provides the lowest possible energy losses, which is an indispensable condition for the operation thrifitness when satisfying several constraints on structural strength of a rocket, stability of flight, and so on. Minimization of energy costs is achieved by reducing the irreversible energy losses on the active flight regime, particularly, losses due to the gravity of the Earth and its atmosphere. These can be expressed as velocity difference when compared with the ideal or characteristic speed the launcher could obtain during a straight flight in the absence of the Earth's gravity field and atmosphere. Choice of a launch best trajectory is based on determining the optimal control law under previously stated limitations, where payload mass is the parameter to be optimized by the differential equations. This trajectory is defined with known basic characteristics of the launch vehicle and given kinematic parameters. Based on above dependencies, it can be noted that maximization of a given payload mass is provided by choosing a law for the angle of attack at which aerodynamic and gravitational losses are reduced and vehicle's \(\Delta V\) is maximized.