The Transiting Exoplanet Survey Satellite (TESS) will search for planets transiting bright and nearby stars. TESS has been selected by NASA for launch in 2017 as an Astrophysics Explorer mission. The spacecraft will be placed into a highly elliptical 13.7-day orbit around the Earth. During its 2-year mission, TESS will employ four wide-field optical charge-coupled device cameras to monitor at least 200,000 main-sequence dwarf stars with I C ≈4−13 for temporary drops in brightness caused by planetary transits. Each star will be observed for an interval ranging from 1 month to 1 year, depending mainly on the star’s ecliptic latitude. The longest observing intervals will be for stars near the ecliptic poles, which are the optimal locations for follow-up observations with the James Webb Space Telescope. Brightness measurements of preselected target stars will be recorded every 2 min, and full frame images will be recorded every 30 min. TESS stars will be 10 to 100 times brighter than those surveyed by the pioneering Kepler mission. This will make TESS planets easier to characterize with follow-up observations. TESS is expected to find more than a thousand planets smaller than Neptune, including dozens that are comparable in size to the Earth. Public data releases will occur every 4 months, inviting immediate community-wide efforts to study the new planets. The TESS legacy will be a catalog of the nearest and brightest stars hosting transiting planets, which will endure as highly favorable targets for detailed investigations.

https://www.spiedigitallibrary.org/journals/Journal-of-Astronomical-Telescopes-Instruments-and-Systems/volume-1/issue-1/014003/Transiting-Exoplanet-Survey-Satellite/10.1117/1.JATIS.1.1.014003.pdf

Transiting Exoplanet Survey Satellite

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L_1 and the other around L_2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L_1 and L_2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the “interior” and “exterior” Hill’s regions and other resonant phenomena.

/pdf/heteroclinic-connections-between-periodic-orbits-and-2j8q5vgrve.pdf

Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics

This paper concerns heteroclinic connections and resonance transitions in the planar circular
restricted 3-body problem, with applications to the dynamics of comets and asteroids and the design
of space missions such as the Genesis Discovery Mission and low energy Earth to Moon transfers.
The existence of a heteroclinic connection between pairs of equal energy periodic orbits around two of
the libration points is shown numerically. This is applied to resonance transition and the construction
of orbits with prescribed itineraries. Invariant manifold structures are relevant for transport between
the interior and exterior Hill’s regions, and other resonant phenomena throughout the solar system.

/pdf/dynamical-systems-the-three-body-problem-and-space-mission-48b7chyrqv.pdf

Dynamical Systems, the Three-Body Problem and Space Mission Design

In 1991, the Japanese Hiten mission used a low energy transfer with a ballistic capture at the Moon which required less ΔV than a standard Hohmann transfer. In this paper, we apply the dynamical systems techniques developed in our earlier work to reproduce systematically a Hiten-like mission. We approximate the Sun—Earth—Moon-spacecraft 4-body system as two 3-body systems. Using the invariant manifold structures of the Lagrange points of the 3-body systems, we are able to construct low energy transfer trajectories from the Earth which execute ballistic capture at the Moon. The techniques used in the design and construction of this trajectory may be applied in many situations.

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Low Energy Transfer to the Moon

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. 
This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

/pdf/connecting-orbits-and-invariant-manifolds-in-the-spatial-2zhs0ys67a.pdf

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

Recently, there has been accelerated interest in missions utilizing trajectories near libration points. The trajectory design issues involved in missions of such complexity go beyond the lack of preliminary baseline trajectories (since conic analysis fails in this region of the solution space). Successful and efficient design of mission options will require new persepectives; a more complete understanding of the solution space is imperative. In this investigation, dynamical systems theory is applied to better understand the geometry of the phase space in the three-body problem via stable and unstable manifolds. Then, the manifolds are used to generate various solution arcs and establish trajectory options that are then utilized in preliminary design for the proposed Suess-Urey mission.

Martin W. Lo

Papers

Heteroclinic Connections Between Periodic Orbits and Resonance Transitions in Celestial Mechanics

Dynamical Systems, the Three-Body Problem and Space Mission Design

Low Energy Transfer to the Moon

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

Application of Dynamical Systems Theory to Trajectory Design for a Libration Point Mission