Patched-conic Approximation Research Articles

An analytical study of the powered swing-by to perform orbital maneuvers

In the present paper, we present an analytical description of the powered swing-by maneuver in the three-dimensional space. To perform this task, new analytical equations are derived for the three-dimensional unpowered swing-by, including the variation in velocity, angular momentum, energy and inclination of the spacecraft due to the maneuver. This is done based on the patched-conics approximation. From those general three-dimensional equations, it is possible to confirm some well-known analytical results for the planar case. The equations developed here are verified by numerical integrations, using the restricted problem of three bodies, showing an agreement better than 1%. After that, based on those equations, some new analytical results are obtained for the situation when an impulse is applied to the spacecraft during the close approach. Those new equations are based in the assumption that the impulse applied is small compared to the velocity of the spacecraft, and they allow the calculation of the same variations cited above. Those equations can be simplified for specific cases, and it can generate some conclusions about the direction to apply the impulse to attend specific goals. A study is also executed to investigate in which cases the impulse is more efficient when applied at the periapsis of the swing-by hyperbola or at points far way from the swing-by body.

Orbital control of a satellite using the gravity of the Moon

In this paper, a study is made in the problem of the orbital control of an Earth´s satellite using the gravity of the Moon. The main objective is to study a technique to decrease the fuel consumption of a plane change maneuver to be performed in a satellite that is in orbit around the Earth. The main idea of this approach is to send the spacecraft to the Moon using a single impulsive maneuver, use the gravity field of the Moon to make the desired plane change of the trajectory, and then return the spacecraft to its nominal semi-major axis and eccentricity using a bi-impulsive Hohmann type maneuver. The spacecraft is assumed to start in a circular orbit in the plane of the lunar orbit around the Earth and the goal is to put it in a similar orbit that differs from the initial orbit only by the inclination. A description of the close approach maneuver is made in the three-dimensional space. Analytical equations based in the patched conics approximation are used to calculate the variation in velocity, angular momentum, energy and inclination of the spacecraft that realizes this maneuver. Then, several simulations are made to evaluate the savings involved.

Classification of Out-of-Plane Swing-By Trajectories

The swing-by maneuver uses a close approach with a celestial body to modify the velocity, energy, and angular momentum of a spacecraft. The literature has several papers studying this problem, usually using a patchedconic approximation. In the present paper the swing-by maneuvers are studied and classiŽ ed under the model given by the three-dimensional restricted three-body problem. To show the results, the orbits of the spacecraft are classiŽ ed in four groups: elliptic direct, elliptic retrograde, hyperbolic direct, and hyperbolic retrograde. Then the modiŽ cation in the orbit of the spacecraft caused by the close approach is shown in plots that specify from which group of orbits the spacecraft is coming and to which group it is going. Several families of orbits are found and shown in detail. The effect of every parameter involved in this problem is studied individually. The results generated here have a potential use to solve optimal problems, such as Ž nding trajectories that satisfy some given constraints (such as achieving an escape or capture) with some parameters being extremized (position, velocity, etc.). Two optimal problems are solved in this paper to show this application.

Minimum-Fuel Escape from Two-Body Sun-Earth System

Escaping from the solar system by receiving gravity assist from the Earth is considered in this paper. A simple procedure,which neglects the eccentricity of the Earth’s orbit and uses the two-body problem equations and the patched-conic approximation, provides near-optimal trajectories using either a single or multiple Earth flybys. The analysis shows that the amount of propellant required to escape from the solar system decreases with the number of flybys, but the mission time increases. The same approach is also used to find near-optimal trajectories that use a single powered flyby. The eccentricity of the Earth’s orbit can be exploited to reduce the characteristic velocity; an indirect optimization procedure provides the most favorable locations where the Earth should be intercepted.

Optimization of ? V Earth-Gravity-Assist Trajectories

An analysis of delta-velocity Earth-gravity-assist (D V-EGA) trajectories is carried out by using an indirect optimization method. In comparison to previous literature, the eccentricity of the Earth’ s orbit is taken into account, and simplifying assumptions, concerning the position and direction of velocity impulses, are removed. However, the focus is on the application of the theory of optimal control to complex interplanetary missions and D V-EGA trajectories are only illustrative examples. In particular, it is shown that the optimization of this kind of mission, according to the patched-conic approximation, can be obtained by considering only the two-body problem equations in the heliocentric frame. The necessary optimum conditions for the Earth e yby are extended to consider a minimum-height constraint, which is frequently required in this maneuver. The procedure is also capable of recognizing when a numerical solution is only suboptimal and a powered e yby could improve the mission.