Nodal Period Research Articles

Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control

This study considers the problem of injecting a spacecraft into an elliptic, repeating-ground-track orbit about Mars, starting from a 4-sol highly elliptical orbit, which is a typical Martian capture orbit, entered at the end of the interplanetary transfer. The final operational orbit has apoares corresponding to the maximum (or minimum) latitude, and nine nodal periods are flown in 5 Martian nodal days. The orbit at hand is proven to guarantee coverage properties similar to the Molniya orbit about Earth; therefore, it is especially suitable for satellites that form constellations. Low-thrust nonlinear orbit control is proposed as an affordable and effective option for orbit injection, capable of attaining significant propellant reduction if compared to alternative strategies based on chemical propulsion. This work introduces a new, saturated feedback law for the low-thrust direction and magnitude that is capable of driving the spacecraft of interest toward the operational orbit. Remarkable stability properties are proven to hold using the Lyapunov stability theory. Because no reference path is to be identified a priori, this technique represents a viable autonomous guidance strategy, even in the case of temporary unavailability of the low-thrust propulsion system or in the presence of widely dispersed initial conditions and errors on estimating orbit perturbations. Monte Carlo simulations prove that the feedback guidance strategy at hand is effective and accurate for injecting a spacecraft into the desired, repeating-ground-track operational orbit without requiring any reference transfer path.

Orbital maneuver technology of temporary networked satellites for earth exploration

It is very difficult to achieve effective coverage of hot-spot target in a short time period by earth perturbation for orbital maneuver, because the coverage gap of the temporary multi-satellites network in certain period of time is too large. In this paper, it is assumed that wave beam type of the satellite-payload sensor is a simple conic, the beam center of the satellite payload under the maximum lateral pendulum is calculated based on the precise ephemeris, and according to the projection ellipse of the payload wave beam on the ground, the coverage ability of the satellite payload on the ground target direction is obtained. Then the satellite is designed to encounter the ground target in the nodal period between the satellite passing the ground target and the latitude circle. The orbital maneuver control variables are calculated from the change of nodal period, and the optimal orbital maneuver control variables are selected from a specified time period. At last, the orbit maneuver in covering the target in a specified time period is achieved and the maximum revisit interval is greatly shortened.

Exciting Mutual Inclination in Planetary Systems with a Distant Stellar Companion: The Case of Kepler-108

Abstract We study the excitation of mutual inclination between planetary orbits by a novel secular-orbital resonance in multi-planet systems perturbed by binary companions, which we call “ivection.” The ivection resonance happens when the nodal precession rate of the planet matches a multiple of the orbital frequency of the binary, and its physical nature is similar to the previously studied evection resonance. Capture into an ivection resonance requires encountering the resonance with slowly increasing nodal precession rate, and it can excite the mutual inclination of the planets without affecting their eccentricities. We discuss the possible outcomes of ivection resonance capture, and we use simulations to illustrate that it is a promising mechanism for producing the mutual inclination in systems where planets have significant mutual inclination but modest eccentricity, such as Kepler-108. We also find an apparent deficit of multi-planet systems that would have a nodal precession period comparable to the binary orbital period, suggesting that ivection resonance may inhibit formation of or destablize multi-planet systems with an external binary companion.

Invariant relative orbits for spacecraft formation flying in high-order gravitational field

This study addresses the boundary and fluctuation of invariant relative orbits for spacecraft formation flying in a high-order gravitational field. The invariant relative condition is analyzed to obtain the analytical boundary. To ensure the accuracy of invariant condition, zonal harmonic terms of the gravitational field, from J 2 to J 15 , are considered. Short- and long-period terms of Delaunay orbital elements are individually eliminated via Hori–Lie transformations. The distribution and evolution of periodic orbits are revealed, along with their corresponding nodal period and drift of the right ascension of the ascending node (RAAN) via single-average Delaunay elements including long-period and secular terms. The corresponding double-average elements with only secular terms are used to match the invariant relative orbits. The boundary and fluctuation of invariant relative motion are derived analytically, which can be treated as new indices for the design of invariant relative orbits. Moreover, the accuracy of the analytical boundary and fluctuation is verified compared with the results of numerical integration. • Analytical invariant relative condition under high-order gravity field. • Analytical boundary and fluctuation of the invariant relative motion. • The influence of the relative elements on relative motion.

Bounded motion design in the Earth zonal problem using differential algebra based normal form methods

Establishing long-term relative bounded motion between orbits in perturbed dynamics is a key challenge in astrodynamics to enable cluster flight with minimum propellant expenditure. In this work, we present an approach that allows for the design of long-term relative bounded motion considering a zonal gravitational model. Entire sets of orbits are obtained via high-order Taylor expansions of Poincare return maps about reference fixed points. The high-order normal form algorithm is used to determine a change in expansion variables of the map into normal form space, in which the phase space behavior is circular and can be easily parameterized by action–angle coordinates. The action–angle representation of the normal form coordinates is then used to parameterize the original Poincare return map and average it over a full phase space revolution by a path integral along the angle parameterization. As a result, the averaged nodal period and drift in the ascending node are obtained, for which the bounded motion conditions are straightforwardly imposed. Sets of highly accurate bounded orbits are obtained, extending over several thousand kilometers, and valid for decades.

Design of Bounded Relative Trajectories in the Earth Zonal Problem

Numerical methods have proven to be reliable tools in generating long-term bounded relative trajectories in nontrivial dynamical environments. Yet, none of the existing procedures have a systematic approach to come up with all of the bounded relative trajectories associated with a variety of satellites in Earth orbit. In this paper, such a systematic procedure is developed based on the assumption that any trajectory, except at critical inclinations, is either periodic or quasi periodic in the four-dimensional Routh reduced system describing the motion of a mass particle about an axisymmetric body. This allows key design parameters to be identified, such as the nodal period and the node drift per nodal period, which are later used to constrain a differential corrector scheme looking for families of quasi-periodic invariant tori that yield bounded relative motion in the Earth-centered inertial frame. Numerical simulations show that the computed solutions are good candidates for cluster flight missions at a large range of altitude and inclination values.

High-precision repeat-groundtrack orbit design and maintenance for Earth observation missions

The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellites. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to $$J_{15}$$ . A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.

Bounded relative motion under zonal harmonics perturbations

The problem of finding natural bounded relative trajectories between the different units of a distributed space system is of great interest to the astrodynamics community. This is because most popular initialization methods still fail to establish long-term bounded relative motion when gravitational perturbations are involved. Recent numerical searches based on dynamical systems theory and ergodic maps have demonstrated that bounded relative trajectories not only exist but may extend up to hundreds of kilometers, i.e., well beyond the reach of currently available techniques. To remedy this, we introduce a novel approach that relies on neither linearized equations nor mean-to-osculating orbit element mappings. The proposed algorithm applies to rotationally symmetric bodies and is based on a numerical method for computing quasi-periodic invariant tori via stroboscopic maps, including extra constraints to fix the average of the nodal period and RAAN drift between two consecutive equatorial plane crossings of the quasi-periodic solutions. In this way, bounded relative trajectories of arbitrary size can be found with great accuracy as long as these are allowed by the natural dynamics and the physical constraints of the system (e.g., the surface of the gravitational attractor). This holds under any number of zonal harmonics perturbations and for arbitrary time intervals as demonstrated by numerical simulations about an Earth-like planet and the highly oblate primary of the binary asteroid (66391) 1999 KW4.

Sea-Level Forcing by Synchronization of 56- and 74-Year Oscillations with the Moon's Nodal Tide on the Northwest European Shelf (Eastern North Sea to Central Baltic Sea)

ABSTRACT Hansen, J.M.; Aagaard, T., and Kuijpers, A., 2015. Sea-level forcing by synchronization of 56- and 74-year oscillations with the Moon's nodal tide on the northwest European Shelf (eastern North Sea to central Baltic Sea). The North Sea and Baltic Sea long-term records reveal a strong correlation (0.997) between sea-level changes and the sum of identified harmonic oscillations, corresponding to the lunar nodal period and four multiples of it. We developed a transparent method for iterative least residual sine regression that is capable of identifying harmonic sea-level oscillations, e.g., gravitational sea-level effects of the lunar nodal oscillation. Three relatively large harmonic sea-level oscillations with period lengths of 18.6 (18.6), 60.5 (55.8), and 76.1 (74.4) years correspond well to factors 1, 3, and 4 of the 18.6-year lunar nodal period (multiple periods in parentheses). The sum of these oscillations leaves small residuals that can be resolved into two further, statistically less signi...

Long-Period Tides in an Atmospherically Driven, Stratified Ocean

AbstractLong-period tides (LPT) are studied using a stratified, primitive equation model on a global domain and in the presence of a fully developed, atmospherically forced ocean general circulation. The major LPT constituents, from termensual to nodal (18.6 yr) periods, are examined. Ocean circulation variability can overwhelm the longest tide signals and make inferring LPT from data difficult, but model results suggest that bottom pressure offers cleaner signal-to-noise ratios than sea level, particularly at low latitudes where atmospherically driven variability is substantially stronger at the surface than at the bottom. Most tides exhibit a significant large-scale dynamic response, with the tendency for weaker nonequilibrium signals in the Atlantic compared to the Pacific as seen in previous studies. However, across most tidal lines, the largest dynamic signals tend to occur in the Arctic and Nordic Seas and also in Hudson Bay. Bathymetry and coastal geometry contribute to the modeled nonequilibrium behavior. Baroclinic effects tend to increase with the tidal period. Apart from short spatial-scale modulations associated with topographic interactions, the excitation of various propagating baroclinic wave modes is clearly part of the modeled LPT, particularly at tropical latitudes, for fortnightly and longer-period tides.

Long-term passive distance-bounded relative motion in the presence of $$J_2$$ J 2 perturbations

This paper presents closed-form solutions for the problem of long-term satellite relative motion in the presence of \(J_2\) perturbations, and introduces a design methodology for long-term passive distance-bounded relative motion. There are two key ingredients of closed-form solutions.One is the model of relative motion; the other is the Hamiltonian model and its canonical solution of the \(J_2\)-perturbed absolute motion. The model of relative motion makes no assumptions on the eccentricity of the reference orbit or on the magnitude of the relative distances. Besides, the relative motion model is concise with straightforward physical insight, and consistent with the Hamiltonian model. The Hamiltonian model takes into account the secular, long-periodic and short-periodic effects of the \(J_2\) perturbation. It also remains separable in terms of spherical coordinates to ensure the application of the Hamilton–Jacobi theory to derive the canonical solution. When deriving the canonical solution, pseudo-circular and pseudo-elliptical orbits are treated separately and Carlson’s method is employed to calculate elliptic integrals, which takes advantage of the symmetry of the integrand. These symmetry properties hold physical insights of the \(J_2\)-perturbed absolute motion. To design the long-term distance-bounded relative motion, the nodal period and the drift of right ascension of the ascending node (RAAN) per nodal period are, respectively, matched non-instantaneously. Even though the nodal period and the drift of RAAN per nodal period can be obtained via the canonical solution, action-angle variables are used to obtain the frequency of the system without finding the complete solution to the perturbed orbital motion.

Treatment Notes: Objective Measures of Language Style Point to Clinical Insights

Application of a computerized text analysis procedure is proposed that has the potential for use by psychoanalytic and psychodynamic clinicians: the systematic examination of linguistic style as reflected by clinicians in their ongoing process and case notes, which are ubiquitous in the mental health field. The studies reported here are, as far as is known, the first attempts to study treatment notes systematically using such procedures. Linguistic measures are used to track the trajectory of the clinical process throughout the treatment in two contrasting cases, one rated successful, the other not. The computerized linguistic analysis used here focuses on two analytically relevant linguistic variables: Mean High Referential Activity (MHW), a measure of the degree to which language is connected to emotional processing, and Reflection (REF), the use of words referring to logical functions. Changes in the relative position of these measures indicate nodal points in the treatment that might be analytically or therapeutically problematic, and that might be overlooked in a solely clinical reading. The analyst's activity as reported in notes during such nodal periods is clinically examined to see how it may have affected the course of the analysis. This method has the potential for use in ongoing treatments, and may help clinicians refine their interventions.

Unique sextuple system: 65 Ursae Majoris

Context. 65 UMa belongs to rather small group of stellar systems of higher multiplicity, whose inner and outer orbits are well-known. Methods: Photometric, spectroscopic, and interferometric data were analyzed, revealing the basic physical properties of the system 65 UMa. A disentangling technique is used to perform the spectra decomposition. This combined approach allows us to study the long-term period changes in the system, identifying the period variation due to the motion on the visual orbit, in addition to a short-term modulation. Results: We find that the system contains one more component, hence 65 UMa is a sextuple hierarchical system. The most inner pair of components consists of an eclipsing binary orbiting around a barycenter on a circular orbit, both components being almost identical of spectral type about A7. This pair orbits on an eccentric orbit around a barycenter, and the third component orbits with a period of about 640 days. This motion is reflected in the period variation in the minima times of the eclipsing pair, as well as in the radial velocities of the primary, secondary, and tertiary components. Moreover, this system orbits around a barycenter with the distant component resolved interferometrically, whose period is of about 118 years. Two more distant components (4" and 63") are also probably gravitationally bound to the system. The nodal period of the eclipsing-pair orbit is on the order of only a few centuries, which makes this system even more interesting for a future prospective detection of changing the depths of minima. Conclusions: We identify a unique solution of the system 65 UMa, decomposing the individual components and even shifting the system to higher multiplicity. The study of this kind of multiple can help us to understand the origin of stellar systems. Besides 65 UMa, only another 11 sextuple systems have been studied.

HS Hydrae about to turn off its eclipses

Aims: We aim to perform the first long-term analysis of the system HS Hya. Methods: We performed an analysis of the long-term evolution of the light curves of the detached eclipsing system HS Hya. Collecting all available photometric data since its discovery, the light curves were analyzed with a special focus on the evolution of system's inclination. Results: We find that the system undergoes a rapid change of inclination. Since its discovery until today the system's inclination changed by more than 15 deg. The shape of the light curve changes, and now the eclipses are almost undetectable. The third distant component of the system is causing the precession of the close orbit, and the nodal period is about 631 yr. Conclusions: New precise observations are desperately needed, preferably this year, because the amplitude of variations is decreasing rapidly every year. We know only 10 such systems on the whole sky at present.

On the existence of J 2 invariant relative orbits from the dynamical system point of view

The present paper addresses the existence of J2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincare techniques in the J2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J2 invariant relative orbits from the dynamical system point of view. A new type of Poincare mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum Hz and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods Td and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincare mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and Hz-contour maps on the space of Td–ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J2 invariant relative orbits to different invariant parameters E and Hz depending on the E- and Hz-contour map, which will hold J2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J2 invariant relative orbits. Therefore, the existence of J2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.

Tidal cycles in the sediments of Santa Barbara Basin

The sediments of Santa Barbara Basin, a valuable source of information about the history of the California Current, show cyclic deposition in three properties here investigated: varve thickness, total organic carbon (TOC) accumulation, and fish-scale abundance. We analyzed the record since A.D. 1350 for varves, A.D. 1120 for TOC, and A.D. 1200 for fish scales. Prominent periods are 169 yr, 93 yr, 51.4 yr, 17.7 yr (varve thickness), 124 yr, 56 yr, 37.4 yr (TOC flux), and 103 yr, 128 yr (fish-scale abundance). A surprising number of the varve and TOC cycles can be interpreted as deriving from the common tidal cycles, i.e., the 4.425 yr perigee period (describing the changing distance between Earth and Moon) and the 18.6 yr lunar nodal period (which pertains to the lineup of Earth, Moon, and Sun). We suggest that the record of the basin is partly a result of pulsed sedimentation aligned with tides. These findings support the hypothesis that a considerable portion of sediment destined for deposition in the center of the basin goes through a phase of shallow storage on the shelf where it can be remobilized through storms and tides. Mechanisms calling on tidally pulsed productivity or tidally influenced climate change are not excluded, but are considered less attractive than remobilization. Fish-scale cycles may reflect the influence of solar forcing.

An Astronomical Basis for the Myth of the Solar Hero

Our increasing knowledge of the megalithic culture of the British Isles in the 2nd and 3rd millennia BCE tends to confirm the proposition that megalithic astronomers measured celestial positions with considerable accuracy. The evidence indicates that they understood the 18.6 year nodal period and the moon’s nine minute declination wobble. They also had sufficient geometrical ability to re-proportion spacings between lines, divide circles into whole number polygons and divide lines into equal integer spacings. We should therefore ask whether there is evidence of such early astronomy in the numbers which recur in certain myths.

Precambrian length of day and the validity of tidal rhythmite paleotidal values

Paleotidal records obtained from sedimentary tidal rhythmites may be systematically abbreviated and so may give incorrect paleotidal and paleorotational values. The validity of determined values, including past length of day (l.o.d.), can be assessed by testing for internal self‐consistency through application of the laws of celestial mechanics. Three independent values obtained from the ∼620‐Ma Elatina‐Reynella rhythmites in South Australia (14.1 sidereal months/year, 401 sidereal days/year, and 19.5 years for the lunar nodal period), when employed in different equations that make allowance for lunar and solar tidal effects, each give a lunar semimajor axis in the range of 96.5–96.9% of the present figure. Such self‐consistency strongly supports the validity of the derived l.o.d. of 21.9 hours at ∼620 Ma. The validity of the estimated l.o.d. of 20.9 hours at ∼900 Ma (revised value, Big Cottonwood rhythmites, Utah) and of 17.1–18.9 hours at ∼2.5 Ga (Weeli Wolli rhythmites, Western Australia) cannot be assessed in that way because each data set has only one directly determined value. The derived mean rate of lunar retreat of 2.16 cm/year since ∼620 Ma averts a close approach of the Moon at least since 3 Ga and a lower rate of retreat seems likely during the Proterozoic.

Orbits in an anisotropic radiation field

AbstractThe elliptic‐type motion around a source whose luminosity is anisotropic is being perturbatively treated. Using a Fourier expansion for the perturbing force, exact expressions describing the evolution of each orbital element are determined from Newton‐Euler equations. For near‐unit frequency the resonant solutions are pointed out. An approximate expression for the variation of the nodal period is determined, too. The behaviour of the radius vector of the unstable orbit (resonance case) is pointed out, showing periodic variations of constantly increasing amplitude. The (both angular and physical) time scale for escape is estimated. Concrete astronomical situations modellable in this way are mentioned.

Drag perturbations in artificial satellite orbital motion based on the Total Density model

Analytical expressions for the aerodynamic drag perturbations undergone by five independent orbital elements of an artificial satellite moving in an orbit with small eccentricity, over a nodal period, are established. All three components of the perturbing acceleration are taken into account. The TD model is used to describe the density distribution of the upper atmosphere.