Reduction of Saturn Orbit Insertion Impulse using Deep-Space Low Thrust

Orbit insertion at Saturn requires a large impulsive manoeuver due to the velocity difference between the spacecraft and the planet. This paper presents a strategy to reduce dramatically the hyperbolic excess speed at Saturn by means of deep-space electric propulsion. The interplanetary trajectory includes a gravity assist at Jupiter, combined with low-thrust maneuvers. The thrust arc from Earth to Jupiter lowers the launch energy requirement, while an ad hoc steering law applied after the Jupiter flyby reduces the hyperbolic excess speed upon arrival at Saturn. This lowers the orbit insertion impulse to the point where capture is possible even with a gravity assist with Titan. The control-law algorithm, the benefits to the mass budget and the main technological aspects are presented and discussed. The simple steering law is compared with a trajectory optimizer to evaluate the quality of the results and possibilities for improvement

orvara::An Efficient Code to Fit Orbits using Radial Velocity, Absolute, and/or Relative Astrometry

We present an open-source Python package, Orbits from Radial Velocity, Absolute, and/or Relative Astrometry (orvara), to fit Keplerian orbits to any combination of radial velocity, relative astrometry, and absolute astrometry data from the Hipparcos-Gaia Catalog of Accelerations. By combining these three data types, one can measure precise masses and sometimes orbital parameters even when the observations cover a small fraction of an orbit. orvara achieves its computational performance with an eccentric anomaly solver five to ten times faster than commonly used approaches, low-level memory management to avoid python overheads, and by analytically marginalizing out parallax, barycenter proper motion, and the instrument-specific radial velocity zero points. Through its integration with the Hipparcos and Gaia intermediate astrometry package htof, orvara can properly account for the epoch astrometry measurements of Hipparcos and the measurement times and scan angles of individual Gaia epochs. We configure orvara with modifiable .ini configuration files tailored to any specific stellar or planetary system. We demonstrate orvara with a case study application to a recently discovered white dwarf/main sequence (WD/MS) system, HD 159062. By adding absolute astrometry to literature RV and relative astrometry data, our comprehensive MCMC analysis improves the precision of HD 159062B's mass by more than an order of magnitude to $0.6083^{+0.0083}_{-0.0073}\,M_\odot$. We also derive a low eccentricity and large semimajor axis, establishing HD 159062AB as a system that did not experience Roche lobe overflow.Comment: 24 pages, 5 figures, 5 tables. AJ accepted with minor changes. orvara is available at https://github.com/t-brandt/orvar

Regular propagators and other techniques in orbit determination problems

The Space Surveillance Network detects and registers Earth-orbiting man-made objects of a size larger than 10 cm, currently accounting for over 24, 000 objects, including nearly 2, 000 active satellites and 3, 000 non-operational satellites, with these numbers going up as new sensing capabilities become available in the coming years. The knowledge of their orbits is of paramount importance for two basic types of problems, namely the propagation and the determination of their orbits. In orbit propagation, the dynamical state of the satellite is known a priori at a given instant, and the aim is to compute the dynamical state at later instants. On the contrary, orbit determination is about estimating the dynamical state of the satellite at a given instant, making use of observation data from one or more tracking stations. Propagation and orbit determination are thus inverse processes and complementary to each other, since the determination process requires the propagation of the orbit, and vice versa. The application of these techniques naturally arises as a fundamental element in the field of space situational awareness, where automated computer systems routinely perform the precise orbit determination of both operational and inactive artificial satellites. Also, operational satellites require the precise knowledge of their orbits to predict the moment they fly over tracking stations, as well as to safely operate active satellites and plan collision avoidance maneuvers. Consequently, these automated systems not only need to be robust and reliable, but due the large amounts of data that need to be processed daily, they also need to be computationally efficient. The aim of this thesis is to develop robust and efficient numerical methods to be applied in both orbit determination and orbit propagation problems. The primary input for orbit determination are observations. For the determination of a Keplerian orbit six independent observations are required, although often many more are available. Thus, preliminary orbit determination is devoted to the estimation of a Keplerian orbit using only six observations. Observational data can be of a varied nature (angles only, range and range-rate, ...), leading to a wealth of wellknown classical techniques, e.g. Gibbs, Herrick-Gibbs, Laplace and Gauss methods. Solving Kepler’s equation is a commonplace to all of them. As a consequence, fast, accurate, robust and reliable methods for solving Kepler’s equation are of a vital importance in preliminary orbit determination. This dissertation proposes one such algorithm that, based on an improved initial guess, succeeds to solve the elliptic and hyperbolic Kepler’s equation more efficiently. Another recurring element in many preliminary orbit determination methods is solving the two-body Lambert’s problem. This dissertation revisits this problem from a new viewpoint, providing an innovative, insightful approach that enables a robust and efficient solution to the problem. Real satellite orbits, however, are not Keplerian due to the many perturbing forces acting upon them, and thus six observations are not enough to realistically capture the dynamical state of the satellite. If more observations are available, a better estimate is possible; this is the focus of statistical orbit determination, that aims to determine the initial state of a satellite from multiple observations and under a realistic, non-Keplerian dynamical model. The corresponding techniques are usually based on a linearization around the outcomes of preliminary orbit determination, leading to a succession of values converging to the sought solution, so that the propagation with the complete dynamical model yields an orbital solution that provides the best possible fit to the actual observations. Consequently, these methods rely on robust, accurate and efficient orbit propagation methods, so the choice of an appropriate propagation technique is fundamental. In this regard, classical methods (e.g. Cowell’s method) are commonly used; these methods, although effective and simple to implement, have the drawback of an exponential error growth and singular equations of motion. Conversely, regularized orbital formulations, such as DROMO, Kustaanheimo-Stiefel, or Sperling-Bürdet, allow to reformulate the equations of motion so that the orbital dynamics is described by a system of non-singular equations, which facilitates the numerical integration and reduces the negative effects of the Lyapunov instability associated to Keplerian motion, thus providing far more accurate and efficient propagation routines. With this vision in mind, the second part of this dissertation is devoted to orbit propagation with regularized formulations, and makes new contributions to this field by developing a new reformulation of DROMO, which is specifically tailored for highly-perturbed dynamical environments. This dissertation is organized as follows: • The first part covers the research contributions to orbit determination. Chapter 1 introduces the basics on orbit determination and describes the classical methods; Chapters 2 and 3 present novel methods for solving the elliptic and hyperbolic Kepler’s equation, respectively; Chapter 4 presents a new formulation, valid for any type of orbit, to solve the two-body Lambert’s problem. • The second part of the dissertation focuses on orbit propagation with regularized formulations. Chapter 5 introduces the basics on regularized special perturbation methods; Chapters 6 and 7 describe in detail the DROMO and ElliDROMO regularized formulations, respectively; Chapter 8 extends the two formulations to enhance their performance in strongly perturbed environments, leading to new methods referred to as DROMO-SPE and ElliDROMOSPE; these propagators are tested in different scenarios and their performance is shown; Chapter 9 explores the application of DROMO and DROMO-SPE to the propagation of meteor trajectories and presents a sensitivity analysis of the parameters considered in the physical modeling. Finally, Chapter 10 summarizes the main conclusion of the thesis and traces potential avenues for future work

An efficient code to solve the Kepler equation. Elliptic case

A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behaviour of the modified Newton?Raphson method when the initial seed is close to the looked for solution. To determine a good initial seed the eccentric anomaly domain [0, ?] is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained by requiring six conditions at both ends of the corresponding interval. Thus the real function and the polynomial have equal values at both ends of the interval. Similarly relations are imposed for the two first derivatives. In the singular corner of the Kepler equation, M smaller than 1 and 1 ? e close to zero an asymptotic expansion is developed. In most of the cases, the seed generated leads to reach machine error accuracy with the modified Newton?Raphson method with no iterations or just one iteration. This approach improves the computational time compared with other methods currently in use

On the resolution of the Lambert's problem with the SDG-code. Part I.

Based on the Lambert`s problem, once identified the one-parameter family of orbits that verify the geometric constraints of the problem, we must express the orbits based on a single parameter that allows to select those that satisfy the kinematic condition. The aim of this paper is to reformulate the problem choosing as parameter the true anomaly of the bisector defined by the directions of the two position vectors. The algorithm applied is the SDG-code, developed by the Space Dynamics Group at UPM, which has already been assessed on the resolution of the Kepler equation proving its stability and reliability

Solving the Kepler equation with the SDG-code

A new code to solve the Kepler equation for elliptic and hyperbolic orbits has been developed. The motivation of the study is the determination of an appropriate seed to initialize the numerical method, considering the optimization already tested of the well known Newton-Raphson method. To do that, we take advantage of the full potential of the symbolic manipulators. The final algorithm is stable, reliable and solves successfully the solution of the Kepler equation in the singular corner (M << 1 and e ~ 1)

On the resolution of the Lambert's problem with the SDG-code. Part I.

Based on the Lambert`s problem, once identified the one-parameter family of orbits that verify the geometric constraints of the problem, we must express the orbits based on a single parameter that allows to select those that satisfy the kinematic condition. The aim of this paper is to reformulate the problem choosing as parameter the true anomaly of the bisector defined by the directions of the two position vectors. The algorithm applied is the SDG-code, developed by the Space Dynamics Group at UPM, which has already been assessed on the resolution of the Kepler equation proving its stability and reliability

Solving the Kepler equation with the SDG-code

A new code to solve the Kepler equation for elliptic and hyperbolic orbits has been developed. The motivation of the study is the determination of an appropriate seed to initialize the numerical method, considering the optimization already tested of the well known Newton-Raphson method. To do that, we take advantage of the full potential of the symbolic manipulators. The final algorithm is stable, reliable and solves successfully the solution of the Kepler equation in the singular corner (M << 1 and e ~ 1)

Gravitational Capture at Saturn with Low-Thrust Assistance

Orbit insertion at Saturn requires a large manoeuver with chemical thrusters to compensate for the velocity difference between the spacecraft and the planet. The impact that this has on the propellant budget is severe. This paper discusses an alternative strategy: after a gravity assist with Jupiter, an electrical motor with an ad hoc thrusting law reshapes the orbit and minimizes the hyperbolic excess speed at Saturn, thus facilitating the capture. The control law algorithm, as well as the dynamical and technological aspects are presented and discussed