## 2 Modeling of Hybrid-Propellant Launch Vehicle
### Environment Model  
The environmental model is an indispensable element for enhancing flight safety, improving operational skills, and optimizing the design and performance of the launch vehicle. For this reason, the influence of the environmental effects on the performance and stability of air and space vehicles is also examined in our analysis. Various factors affect different types of vehicles but, for launch vehicles, the atmosphere and the Earth's gravity field are the dominant factors. Therefore, these two models are specifically considered in this study.  
The gravity model plays a crucial role in flight simulation because flight involves understanding the movement of objects within the atmosphere based on these fundamental physical principles. During flight simulation, gravity constitutes the weight of the aircraft, and this weight is overcome by thrust to achieve the launch of the vehicle. _The EGM2008 gravity model_ is a recent global high-degree potential model that offers significant advancements in obtaining precise trajectories for aerospace vehicles [41]. It simplifies the gravitational potential function to derive the gravitational acceleration function (GAF). The equation representing the GAF concerning Earth-centered Earth-fixed (ECEF) coordinates is given as Equation (1) [42]:  
\[G_{l_{2}}^{ECEF}=-\frac{3}{2}I_{2}\Big{(}\frac{\mu}{r^{2}}\Big{)}\Big{(}\frac{ a}{r}\Big{)}^{2}\begin{bmatrix}\left(1-5\left(\frac{z}{r}\right)^{2}\right)& \frac{x}{r}\\ \left(1-5\left(\frac{z}{r}\right)^{2}\right)&\frac{y}{r}\\ \left(1-5\left(\frac{z}{r}\right)^{2}\right)&\frac{z}{r}\end{bmatrix} \tag{1}\]  
where \(J_{2}\) represents the zonal harmonic coefficient (1.08263 \(\times\) 10\({}^{-3}\)), \(\mu\) represents Earth's gravitational constant (3.986 \(\times\) 10\({}^{14}\) m\({}^{3}\)/s\({}^{2}\)), \(r\) represents the distance of the launch vehicle from the center of mass of Earth, \(a\) represents the semi-major axis of Earth (6,378,137 m), and \(x\), \(y\), \(z\) represent the ECEF position of the launch vehicle.  
The atmospheric model is used to predict the flight performance of launch vehicles and understand aerodynamic effects. The atmospheric model defines atmospheric parameters such as air density, pressure, and temperature based on the altitude of the launch vehicle. This information forms the basis for calculating significant forces in flight simulation, such as gravity, aerodynamic forces, and thrust. Consequently, flight simulations more accurately predict real flight conditions and play a crucial role in the design and optimization of launch vehicles. The _US1976 Standard Atmosphere Model (SAM)_ was chosen for this study due to its combination of accuracy and simplicity [43]. This model provides the vertical distribution of atmospheric characteristics based solely on altitude (\(h\)). The outputs obtained from the US1976 SAM include temperature, pressure, speed of sound (\(T_{th},P,Cs\)), and density (\(\rho\)). From these parameters, the M and dynamic pressure (\(P_{dyn}\)) can be derived. For \(h>84\) km, the atmospheric data are obtained analytically, where the tables are used for higher \(h\)[43]. The equations for \(h>84\) km and atmospheric data for higher \(h\) were implemented in the MATLAB environment using the research of NOAA, NASA, and USAF [43].