# A Brazilian Space Launch System for the Small Satellite Market
## 2 Methodology and Launch Vehicle Configuration  
### Vehicle Performance  
For vehicle performance predictions, we used the _DBallistic_ model developed by the Dnepropetrovsk National University (DNU) from Ukraine. The algorithm is based on a simple two-dimensional trajectory analysis. The accuracy of the model is in the range of 0.5 to 3.0%, considered adequate for the conceptual design phase carried out in this work. Essentially, the _DBallistic_ model solves the reverse orbital mechanic's problem with payload mass as the main criteria for trajectory optimization [12; 13]. The task is solved from numerical analysis of the launch vehicle equations of motion. The system of equation is based on the hypothesis of a non-rotating Earth. Figure 3 shows the reference coordinate system and parameters for the trajectory analysis. Within the active phase of the trajectory of the i-th stage of the rocket, the system of equations is integrated with respect to the following dimensionless variable  
\[\mu_{i}=\dot{m_{i}}\left(t-t_{k,i-1}\right)/m_{0i} \tag{1}\]  
For a Cartesian earth-fixed coordinate frame, as shown in Figure 4, the planar rocket trajectory can be calculated by a set of differential equations with \(\mu_{i}\) as the independent variable [14]. Acceleration and velocity along the orthogonal \(X_{a}\)-\(Y_{a}\) plane (Figure 3) are given by  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]  
\[\frac{dV_{y}}{d\mu_{i}}=A_{1}\sin\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{y}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{x}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}(R+y) \tag{3}\]  
\[\frac{dx}{d\mu_{i}}=A_{3}V_{x} \tag{4}\]  
\[\frac{dy}{d\mu_{i}}=A_{3}V_{y} \tag{5}\]terms \(A_{1}\), \(A_{2}\), and \(A_{3}\) in Equations (2)-(5) are estimated from  
\[A_{1}=\frac{g_{0}}{1-\mu_{i}}\left(I_{sp,0i}+\Delta I_{sp,i}\right)\left[1-\pi \left(h\right)\right] \tag{6}\]  
\[A_{2}=g_{0}\frac{70952}{I_{sp,i}}A_{3}\frac{M^{2}\pi(h)}{1-\mu_{i}}; \tag{7}\]  
\[A_{3}=\lambda_{01}I_{sp,0i} \tag{8}\]  
\[\pi\left(h\right)=p\left(h\right)/p_{0} \tag{9}\]  
where,  
\(M\)--Mach number;  
\(I_{sp,0i}\)--sea-level specific impulse of i-th stage;  
\(\Delta I_{sp,i}\)--change in specific impulse due to the ambient pressure;  
\(\pi\)--ratio of local atmospheric pressure to sea-level pressure as a function of altitude  
\(\theta\)--pitch angle;  
\(\alpha\)--angle of attack. The increase in engine-specific impulse due to altitude gain, on the active propulsion stage, is calculated with the help of the coefficient \(K_{p}\) which varies progressively for the first stage and is null for the second and upper stages. The variation in rocket specific impulse is then given by  
\[\Delta I_{sp,i}=I_{sp,i}-I_{sp,0i}=K_{p}I_{sp,0i} \tag{10}\]  
Rocket angle of attack and the programmed pitch angle variation are given by  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]0  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]1  
\[\frac{dV_{x}}{d\mu_{i}}=A_{1}\cos\left(\vartheta\right)-A_{2}c_{x}\left(M \right)\frac{V_{x}}{V}-A_{2}c_{y}\left(M\right)a\frac{V_{y}}{V}-A_{3}\frac{g_ {0}R^{2}}{r^{3}}x \tag{2}\]2 Schematics of the rocket flight on the active flight regime with respect to the Earth’s coordinate system.