# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 1 Introduction  
The number of microsatellites to be orbited is increasing rapidly, in particular for the delivery and replacement of microsatellites as part of mega constellation programs. This is stimulating a new wave of dedicated launch vehicles capable of offering responsive, flexible and cost-effective services to this huge and new market. Some of these \"micro-launchers\" are listed in Table 1. The mass of payload \(m_{pay}\) to be inserted into LEO is below 1000 kg. In fact, \(m_{pay}\) is the driving element of a space rocket and it is important to define it at the first step of a launcher design. A very popular method of computation of \(m_{pay}\) is offered by the Tziolkowski formula. However, the results obtained using this method are not accurate since the space environment (gravity and aerodynamic forces) and the guidance (direction of thrust) are not taken into account. On the other hand, the accurate computation of a launcher trajectory and the evaluation of launcher performances is a typical and complex problem in the optimization of aerospace trajectories. Launcher performances are evaluated maximizing the final mass that can be set into orbits with different parameters (semimajor axis, eccentricity, inclination). The problem has initial- and final-state variable constraints, path constraints and discontinuities in the mass variation, due to stage separation. This optimization problem can be approached in different ways: all the approaches require educated guesses to initialize some iterative algorithm able to refine the initial guesses and to achieve the optimal solution. In the scientific literature, some papers [1; 2; 3; 4; 5; 6; 7] address the problem of optimizing the ascent trajectory of launch vehicles through indirect approaches. Examples are the multiple-subarc gradient restoration algorithm, proposed by [PERSON], and multiple shooting techniques [5; 6; 7]. The previously cited works [2; 3; 4; 5; 6; 7] require a considerable deal of effort for deriving the analytical conditions for optimality and for the subsequent programming and debugging. Furthermore, these methods can suffer from a slow rate of convergence and an excessive dependence on the starting guess. Alternative approaches [8; 9] are direct in nature, often are more robust, but require discretization of the problem (e.g., through collocation [8; 9]) and the use of dedicated algorithms for nonlinear programming, without avoiding the need of a starting guess. Most recently, the indirect heuristic method [10] was proposed as an effective approach for trajectory optimization of ascent vehicles, with the remarkable feature of not requiring any starting guess. For the purpose of preliminary analysis of the performance of new ascent vehicles, fast, approximate approaches that are exempt from convergence issues and minimize the computational effort are desirable. With this intent, [PERSON] and [PERSON] [11] and [PERSON] et al. [12; 13] propose two numerical approaches for performance evaluation of multistage launch vehicles that aim at obtaining accurate predictions of the performance attainable from a launch vehicle. The aim of this paper is to derive formulas, possibly as simple as the Tziolkowski one, able to obtain in a few seconds the performance of a multistage space rockets and also able to offer trajectory parameters useful as input for the convergence of complex numerical programs with accurate representation of the rocket system and its space environment. The formulas originate from previous work [14], which has been generalized to multistage rockets in [15]; all the rocket state variables are derived in [16]. In the present paper, the method is extended to include the optimal choice of a cost arc and the guidance of the last stage of the launcher till the injection into the selected orbit, so the performance of a space launcher is obtained.