## 3 The Carrier-Launcher System Performance: Preliminary Analyses and Results
### Choice of Engines Needed to Perform the Parabolic Manoeuvre  
The engines adopted in the preliminary configuration are four Rolls-Royce UltraFan engines [56]. These engines satisfy the requirements in terms of thrust to perform the first phase of the launcher release manoeuvre. In the present case, from an altitude of about 6000 m, with a speed of 260 m/s, and M = 0.822, the launcher should perform a pull-up manoeuvre. It should climb to an angle \(\gamma=45^{\circ}\) and up to an altitude of about 8000 m to release the launcher at the beginning of the parabolic flight phase. Figure 15 shows a typical trajectory flown during a parabolic flight [10]. During the \(1.8\,g\) phase of flight (pull-up manoeuvre), the lift (\(L\)) is  
\[L=(1/2)\rho S(V)^{2}C_{L\_climb}=1.8W_{rad}, \tag{12}\]  
and the requested thrust \(T_{req}\) is calculated, considering no acceleration along the climbing trajectory, by  
\[T_{req}=D+W_{red}sen(\gamma)=(1/2)\rho S(V)^{2}C_{D\_climb}+W_{red}sen(\gamma), \tag{13}\]  
where \(\rho\) is the air density, \(S\) the reference surface of the wing, \(V=260\) m/s the velocity, \(D\) the aerodynamic drag, \(W_{red}\) the reduced weight of the carrier configuration at the beginning of the pull-up maneuver, \(\gamma\) is the trajectory angle (the velocity vector and thrust axis are aligned with the trajectory), and \(C_{L\_climb}\) and \(C_{D\_climb}\) are the aerodynamic coefficients of lift and drag of the carrier-launcher configuration, respectively. To evaluate \(W_{red}\) at the beginning of the manoeuvre, we assume that the mass of the remaining fuel is 60% of its initial mass; therefore, the mass of fuel is \(37,300\) kg and the actual reduced weight of the carrier-launcher configuration with four engines is \(W_{red}=952,590\) N.  
In this preliminary analysis, to obtain coherent results, we assume that the carrier starts the manoeuvre depicted in Figure 15 [10] at an altitude of \(h=6000\) m, and while the carrier is climbing, it can safely reach a maximum altitude of about \(h=8000\) m (as a reference, the final value of the trajectory radius is \(R=6300\) m). Therefore, to perform a preliminary verification of the thrust requirements, we refer to the altitude \(h=8000\) m: in this case, the air density is \(\rho=0.52517\) kg/m\({}^{3}\) and from Equation (12) we estimate a value of the lift coefficient \(C_{L\_climb}=0.32\) (very similar to the lift coefficient in cruise condition). The same result was obtained with the use of the AVL software, as previously mentioned. Figure 15: Trajectory flown during a parabolic flight [10]. Equation (13) allows us to estimate the maximum required thrust to execute the manoeuvre. Again, from AVL calculation \(C_{D0\_carrier}=0.013\), and, as performed for the cruise condition, we increase this value by 20% to account for the launcher during the climbing phase of the parabolic manoeuvre, thus \(C_{D0\_climb}=0.0156\). In this case, for a stable flight condition, we must rotate the central flap surfaces (upward direction) of an angle of approximately \(\delta=19.84\) deg, and, in the equilibrium condition, the angle of attack is \(\alpha_{w}=5.24\) deg. These results were obtained with AVL setting the pitching moment equal to zero. The induced drag produces an increase in the drag coefficient of about (AVL result) \(C_{D\_climb}=0.0145\), thus the final value of the manoeuvre drag coefficient is  
\[C_{D\_climb}=C_{D0\_climb}+C_{D\_climb\_ind}=0.0156+0.0145=0.0301. \tag{14}\]  
Inserting this value in Equation (13) we obtain  
\[T_{req}=\left[(1/2)\times 0.52517\times 304.36\times(260)^{2}\times 0.0301+95 2,590\times 0.707\right]/1000=836.4\text{ kN} \tag{15}\]  
and we must compare this value with the available thrust. The PW4000-112 engine [41], considered as a candidate engine in first instance, develops a maximum thrust at a sea level of about \(98,000\) pounds, i.e., \(435.9\text{ kN}\).