## 2 Methodology
### Aircraft Model  
An aircraft state-space model [27] was developed and used as a platform to assess the performance of candidate trajectories. In order to cut down on the simulation's computational intensity, the simplest possible representation was selected, comprising only four states \((h,V,\gamma,m)\), directly related to the intended climb performance studies. The resulting model can be generically described as a single-Degree-of-Freedom (DOF) navigation, two-DOF point mass aircraft state-space model, augmented with a mass \((m)\) state so as to account for engine fuel burn. The exact formulas for the aircraft state equations are given in Equations (1)-(6), expressed with respect to an earth coordinate system.  
Modeling was based upon published aerodynamic and mass data for the F-4 aircraft [23; 24], combined with Cranfield University's in-house gas turbine performance code, Turbomatch, which was used to construct models for the aircraft's engines, outputting Thrust (T), air Mass Flow (MF), Nozzle Pressure Ratio (NPR) and Specific Fuel Consumption (SFC). Turbomatch is a software based Gas Turbine performance simulation tool developed by the Propulsion Engineering Centre at Cranfield University [28]. The tool is a 0-D performance simulation code, featuring OD and transient simulation as well [29]. Turbomatch comprises several pre-programmed modules, which correspond to thermodynamic models of components. They are called up to evaluate the engine output, i.e., thrust or power, specific fuel consumption, etc. Its modularity, which is supported by the implementation of generic component maps, enables the detailed design of any gas turbine configuration. Inlet pressure recovery was modeled as a function of Mach number, as per MIL-E-5007D [30]. Throttle-dependent forces were also included to account for installation effects: The experimental data of References [31; 32] were used to construct surrogate models for spillage and afterbody drag respectively. A schematic representation of the general model arrangement is depicted in Figure 1.  
For the simulation runs, a constant throttle setting was assumed, in accordance with standard practice in aircraft climb sequences [24; 33]. A variable-throttle approach (one that considers the sequence of throttle inputs as a problem variable), as adopted in other studies, would be computationally demanding for the detailed engine representation required for aircraft-engine integration applications, since engine transient response would need to be modeled and is anyway impractical for a real-world scenario, being too complicated to be executed by a human pilot. Flight path angle control was used to control the aircraft's climb rate and airspeed to fly commanded paths in the Pressure Altitude (h)-Mach Number (M) plane. The exact guidance logic employed is addressed in Section 2.3. Flight path angle rate saturation was implemented to the model to represent the aircraft's maximum lift capability and structural strength (Equations (3) and (5))  
\[\dot{h}=V\sin\gamma, \tag{1}\]  
\[\dot{V}=\frac{T-D}{m}-g\sin\gamma \tag{2}\]  
Figure 1: Schematic representation of the aircraft–engine model structure.  
\[\dot{\gamma}=\left\{\begin{array}{c}\min\left\{\frac{\gamma_{c}-\gamma}{\tau}, \ \frac{L_{\max}}{mV}-\frac{g\cos\gamma}{V}\right\}\gamma_{c}>\gamma\\ \max\left\{\frac{\gamma_{c}-\gamma}{\tau},\ \frac{-L_{\max}}{mV}-\frac{g\cos\gamma}{V} \right\}\gamma_{c}<\gamma\end{array}\right. \tag{3}\]  
\[\dot{m}=-SFC\times T \tag{4}\]  
\[L_{\max}=\min\left\{\frac{1}{2}\rho SC_{L_{\max}}V^{2},\ n_{z_{\max}}mg\right\} \tag{5}\]  
\[D=\frac{1}{2}\rho SV^{2}\left(C_{D_{0}}+kC_{L}^{2}\right) \tag{6}\]