## 1 Introduction  
The transfer between two circular, coplanar Keplerian orbits of a spacecraft equipped with a continuous thrust propulsion system is a classic astrodynamics problem [1] that is typically solved in an optimal framework by maximizing (or minimizing) a given scalar performance index as, for example, the propellant consumption or the total flight time. In this context, the optimization procedure described in the fundamental textbook by [PERSON] and [PERSON] [2] is a typical example.  
In fact, when the optimization problem is approached using the well-known indirect method [3; 4], i.e., a method based on the classical theory of the calculus of variations [5], the solution requires the evaluation of the initial value of a set of costate variables, which is usually obtained by numerically solving a two-point boundary value problem (TPBVP). The computation time required by the numerical solution (and also by the convergence of the entire procedure) of the TPBVP associated with the optimization procedure is strongly dependent on the guess of unknown initial costates [6; 7].  
The purpose of this paper is to propose an analytical procedure to accurately approximate the initial costate variables in a coplanar, circle-to-circle minimum-time transfer, in order to initialize the numerical procedure that solves the TPBVP with an initial guess that is close to the actual solution. In particular, this paper considers a mission scenario in which the thrust vector is freely steerable during flight, and the magnitude of the propulsive acceleration varies over a finite range. The proposed procedure, which is consistent with the results from the classical literature [8; 9; 10], is suitable for the case of a propulsive acceleration vector with a low (or very low) maximum magnitude. In this scenario, the minimum-time transfer trajectory can be approximated through a tight spiral [11; 12], and some characteristics of the spacecraft osculating orbit can be described in an analytical (simple) form.  
The proposed procedure is used here to analyze the optimal transfer in a set of both heliocentric and geocentric mission scenarios, as a function of the propulsive acceleration maximum magnitude. In particular, optimal circle-to-circle (rapid) spacecraft trajectories are studied in a classical LEO-GEO transfer and in a rendezvous mission towards Venus, Mars, Jupiter, and comet 29P/Schwassmann-Wachmann 1. Finally, the method discussed in this paper can also be used to rapidly obtain a set of accurate information about the optimal transfer trajectory in order to initialize more refined (trajectory) design procedures, such as those recently proposed by [PERSON] et al. [13].  
A natural extension of the method proposed in this work is to consider the dynamics of a spacecraft, which include the change in mass due to propellant consumption. This important aspect of the thrust vector model introduces a further parameter in the (optimal) design of the mission, namely the specific impulse, the value of which influences the overall transfer performance. On the other hand, the proposed method can be adapted to a scenario based on the use of a propellantless propulsion system such as, for example, a scenario involving a photonic solar sail [14; 15] or [PERSON]'s Electric Solar Wind Sail [16; 17].