# Aerospace
## 2 Mathematical Models  
### Atmospheric Flight Mechanics  
The general kinematic and dynamic equations of motion for atmospheric flight over a spherical planet in a planet-fixed reference frame are summarized below [26]. \[\frac{dr}{dt}=V\sin\gamma \tag{1}\]  
\[\frac{d\theta}{dt}=\frac{V\cos\gamma\cos\chi}{r\cos\phi} \tag{2}\]  
\[\frac{d\phi}{dt}=\frac{V\cos\gamma\sin\chi}{r} \tag{3}\]  
\[\frac{dV}{dt}=-\frac{D}{m}-g\sin\gamma+\omega^{2}r\cos\phi(\sin\gamma\cos\phi- \cos\gamma\sin\phi\sin\chi) \tag{4}\]  
\[V\frac{d\gamma}{dt}=\frac{L}{m}\cos\sigma-g\cos\gamma+\frac{V^{2}}{r}\cos \gamma+2\omega V\cos\phi\cos\chi \tag{5}\]  
\[+\omega^{2}r\cos\phi(\cos\gamma\cos\phi+\sin\gamma\sin\phi\sin\chi)\]\[ \begin{split} V\frac{d\chi}{dt}&=\frac{1}{m}\frac{L\sin \sigma}{cos\gamma}-\frac{V^{2}}{r}\cos\gamma\cos\chi\tan\phi+2\omega V(\tan \gamma\cos\phi\sin\chi-\sin\phi)\\ &-\frac{\omega^{2}r}{\cos\gamma}\sin\phi\cos\phi\ cos\chi\end{split}  \tag{6}\]  
where \(r\) is the radial distance from the center of the planet, \(\theta\) is the longitude, \(\phi\) is the latitude, \(V\) is the velocity, \(\gamma\) is the flight-path angle, and \(\chi\) is the heading angle [26]. The lift force \(L\) and drag force \(D\) are given by  
\[L=qAC_{L},\qquad\qquad D=qAC_{D} \tag{7}\]  
where \(q=\frac{1}{2}\rho V^{2}\) is the dynamic pressure, \(A\) is the vehicle aerodynamic reference area, \(m\) is the vehicle mass, \(C_{L}\) is the lift coefficient, and \(C_{D}\) is the drag coefficient. \(\sigma\) indicates the bank angle, \(g\) is the local gravitational acceleration, and \(\omega\) is the planet rotation rate. Thus the aerocapture vehicle state during atmospheric flight is defined by the vector [\(r\), \(\theta\), \(\phi\), \(V\), \(\gamma\), \(\chi\)]. As shown in Figure 3, the entry flight path angle should be within the aerocapture corridor bounded by [\(\gamma_{\min}\), \(\gamma_{\max}\)]. The width of the corridor is the Theoretical Corridor Width (TCW) defined as \(\gamma_{\min}-\gamma_{\max}\). Given an entry state with all parameters except \(\gamma\), a bisection algorithm implemented in the Aerocapture Mission Analysis Tool (AMAT) is used to successively propagate entry trajectories for an upper and lower bound guess for \(\gamma\) to compute \(\gamma_{\min}\) and \(\gamma_{\max}\) for a drag modulation vehicle design [27]. Once the corridor is computed, aerocapture trajectories can be propagated using the entry state [\(r_{e}\), \(\theta_{e}\), \(\phi_{e}\), \(V_{e}\), \(\gamma_{e}\), \(\chi_{e}\)] till atmospheric exit with a guidance scheme used to determine the drag skirt jettison time in order to achieve the desired apoapsis. The convective heating rate is estimated using the Sutton-Graves empirical relation [28]:  
\[\dot{q}_{c}=K\bigg{(}\frac{\rho_{\infty}}{R_{N}}\bigg{)}^{0.5}V^{3} \tag{8}\]  
where \(\dot{q}_{c}\) is the stagnation-point convective heating rate in W/cm\({}^{2}\), \(K=1.8980\times 10^{-8}\) for both Mars and Venus, \(\rho_{\infty}\) is the freestream air density, and \(R_{N}\) is the vehicle nose radius [29]. The radiative heating for Mars is assumed negligible for the low-ballistic coefficient entry system. This is a limitation of the current study because no empirical relations are readily available, and future studies are recommended to quantify the radiative heating contribution for low-ballistic coefficient entry at Mars. The radiative heating rate for Venus is given by [30; 31]:  
\[\dot{q}_{r}= \begin{cases}K_{1}\rho_{\infty}^{1.2}V^{10.0}R_{N}^{0.49}&\text{ if }V<8000\text{ m per s}\\ K_{2}\rho_{\infty}^{1.2}V^{5.5}R_{N}^{0.49}&\text{if }8000\leq V\leq 10,000 \text{ m per s}\\ K_{3}\rho_{\infty}^{1.2}V^{13.4}R_{N}^{0.49}&\text{if }10,000\leq V\leq 12,000 \text{ m per s}\end{cases}  \tag{9}\]  
where \(\dot{q}_{r}\) is the stagnation point radiative heat flux in W/cm\({}^{2}\), \(K_{1}=3.33\times 10^{-34}\), \(K_{2}=1.22\times 10^{-16}\) and \(K_{3}=3.07\times 10^{-48}\).