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Exploration of the outer planets has experienced new interest with the launch of the Cassini and the New Horizons Missions. At the present time, new technologies are under study for the better use of electric propulsion system in deep space missions. In the present paper, the method of the transporting trajectory is used to study this problem. This approximated method for the flight optimization with power-limited low thrust is based on the linearization of the motion of a spacecraft near a keplerian orbit that is close to the transfer trajectory. With the goal of maximizing the mass to be delivered in Saturn, several transfers were studied using nuclear, radioisotopic and solar electric propulsion systems.

The first mission to Saturn was Pioneer 11 that was launched on April 5, 1973. After making a flyby in Jupiter it determined the mass of the Jupiter's moon Callisto. Looping high above the ecliptic plane and across the Solar System, Pioneer 11 raced toward its appointment with Saturn on September 1, 1979. Voyager 2 was launched on August 20, 1977, and Voyager 1 was launched on September 5, 1977. Voyager 1 reached Saturn on November 12, 1980, followed by Voyager 2 in August 25, 1981. Later, the Cassini-Huygens spacecraft was launched on October 15, 1997. Using the gravity assist technique with the combination Venus-Venus-Earth and Jupiter the spacecraft increased its velocity to a level high enough to reach its final destination. On July 1, 2004, the Cassini-Huygens spacecraft fired its main engine to reduce its speed, allowing the spacecraft to be captured by Saturn's gravity and enter orbit. In May 28, 2008, the Cassini spacecraft passed by Saturn's moon Titan and made its last flyby of the original four-year tour, but Cassini's exploration of Saturn will continue for two more years.

With the advent of both the Deep Space 1 mission (Brophy and Noca [

The performance of a one-stage propulsion system can be roughly characterized by two variables: the maximum ejection velocity, which can be related with the maximum specific impulse and the ratio between the maximum thrust and the engine weight on the ground.

Concerning these two parameters (Marec [

Concerning the operating domain, the propulsion system can be essentially classified into constant ejection velocity (CEV) limited thrust systems, and limited power (LP) variable ejection velocity systems.

For the case of the idealized LP system, the only constraint concerns to the power (

In this paper we studied the electric propulsion system of low-thrust type with limited power. Besides, the low-thrust transfers that are studied here have no constraint on the thrust direction. In general, these constraints can be caused by peculiarities of the attitude control system and the mode of the stabilization of the spacecraft.

A description of the mathematical models used to study the low-thrust transfers is now made, in order to explain the procedures used in the present paper.

The electric propulsion low-thrust system uses the ionization of a propellant and its subsequent acceleration in an electrostatic field or electromagnetic to generate thrust. In systems with chemical propulsion, the ignition of the engine can run for several minutes, but in the case of the systems that use the electric propulsion, the ignition of the engine needs to run for longer times, up to several months in some cases.

The equations of motion are

This approximated method of optimization of flight with ideally controlled small thrust is based on the linearization of the motion of a spacecraft near some reference Keplerian orbit (transporting trajectory). The equation of motion of the spacecraft is

The Hamiltonian for the linearized problem is

More details of the analysis of the trajectories using low thrust and the transporting trajectory method can be found in Beletsky and Egorov [

A power-limited low-thrust transfer to Saturn is considered in this paper. For all cases considered below it is assumed that the spacecraft leaves the Earth's sphere of influence with variable velocity and approaches Saturn with zero velocity. Several analyses were considered as a function of the ratio between the final and initial mass, time of flight (TOF), velocity at infinity, and effective specific power. The unit of power used in this paper is the effective specific power that is [

Nuclear Electric Propulsion (NEP) uses a reactor power system to provide the electricity for thrusters that ionize and accelerate propellant to produce thrust. The application of nuclear systems in space has advantage in situations where the distances from the Sun are large and the solar power density is too low

In this section, the study of the NEP system for a trip to Saturn is performed. Figure

Trajectory of the flight Earth-Saturn with effective specific power (NEP) of 50 W/kg and

Curves for NEP as a function of the ratio

When the TOF is larger, the propellant consumption is smaller. These cases showed that the NEP is a good way to transport large payloads to Saturn. Considering the cases with

The angle of the thrust with the spacecraft velocity is shown in Figure

(a) Angle to the spacecraft velocity (continuous line) and angle to the orbital plane (discontinuous line). (b) Thrust vector (continuous line) and

The behavior of the thrust is presented in Figure

The Deep Space 1 mission used a radioisotope thermoelectric generator that was combined with off-the-shelf ion propulsion systems. This combination provides a combined specific mass of almost

Trajectory of the flight Earth-Saturn with effective specific power (REP) of

In this section, some simulations for the REP system are shown. Figure

Curves for REP as a function of the ratio

The angle of the thrust with the spacecraft velocity is shown in Figure

(a) Angle to the spacraft velocity (continuous line) and angle to the orbital plane (discontinuous line). (b) Thrust vector (continuous line) and

The behavior of the thrust is presented in Figure

In October 1998, NASA launched the Deep Space 1 that was the first interplanetary mission to be propelled by solar electric propulsion (Rayman and Williams [

It is known that the weakness of the SEP technology is the low levels of acceleration that it provides and in the reduced solar irradiance available for photovoltaic power generation at the outer reaches of the solar system. Nevertheless, these drawbacks can be avoided by a suitable design that allows the SEP system to operate efficiently for long periods using a wide range of input powers (Mengali and Quarta [

Here, we studied the possible advantages of SEP to reach Saturn. Figure

Trajectory for the flight Earth-Saturn with effective specific power (SEP) of

Figure

Curves for SEP as a function of the ratio

The angle of the thrust with respect of the spacecraft velocity is shown in Figure

(a) Angle to the spacraft velocity (continuous line) and angle to the orbital plane (discontinuous line). (b) Thrust vector (continuous line) and

The behavior of the thrust is presented in Figure

Low-thrust transfers of the limited power type were considered in this paper. The method of the transporting trajectory was used with the reference orbit composed by a set of short arcs of the keplerian orbits, while the transfer trajectory is subjected to low thrust. Since a maximum power provides a minimum propellant consumption, our goal was to maximize the

thrust

gravatational acceleration

specific impulse

current spacecraft mass

ratio between the final and initial mass

propellet mass consumed by

mass flow rate

Lawden’s primer vector

initial and final times

velocity at infinity

electric power

effective power

acceleration vector

power efficiency (constant).

The authors are grateful to the Foundation to Support Research in the São Paulo State, Brazil (FAPESP) for the research grant received under Contract 2008/10236-3 and 2007/04232-2.