Non-circular Orbits Research Articles

3D image acquisition in scoliosis correction surgery: Radiation exposure of a paediatric patient cohort

3D spine imaging has become a popular technique in orthopaedic surgery to assist with the placement of screws and rods for the treatment of scoliosis. It is most often used in combination with navigation software to reduce the risk of surgical-induced patient paralysis. One way to acquire this type of imaging is to use an X-ray c-arm that rotates around the patient, mimicking a cone beam CT acquisition. Although this technique has been successfully implemented in many centres, there is a deficit of information in the literature with regard to the consequent patient radiation exposure, in particular, where an eccentric or non-circular orbit is utilised. This information is particularly important for paediatric patients, as they are the most likely candidates for this type of surgery. Our study estimates the radiation dose to paediatric patients who have undergone scoliosis correction surgery, using the proprietary non-circular orbit employed by a Ziehm Vision RFD 3D C-arm. We estimate and compare patient organ doses using this eccentric orbit with one where a simple circular orbit is assumed. We further compare our calculated organ doses with those imparted by traditional 2D fluoroscopic techniques, and calculate the risk of future radiation induced cancer for this patient cohort. The doses presented here will be useful to other centres in establishing diagnostic reference levels for this increasingly common technique.

X-ray binaries with neutron stars at different accretion stages

Abstract. Different accretion regimes onto magnetized NSs in HMXBs are considered: wind-fed supersonic (Bondi) regime at high accretion rates <math/> g s-1, subsonic settling regime at lower <math/> and supercritical disc accretion during Roche lobe overflow. In wind-fed stage, NSs in HMXBs reach equilibrium spin periods P* proportional to binary orbital period Pb. At supercritical accretion stage, the system may appear as a pulsating ULX. Population synthesis of Galactic HMXBs using standard assumptions on the binary evolution and NS formation is presented. Comparison of the model P* – Pb (the Corbet diagram), P* – Lx and Pb – Lx distributions with those for the observed HMXBs (including Be X-ray binaries) and pulsating ULXs suggests the importance of the reduction of P* in non-circular orbits, explaining the location of Be X-ray binaries in the model Corbet diagram, and the universal parameters of pulsating ULXs depending only on the NS magnetic fields.

Non-iterative angles-only initial relative orbit determination with [formula omitted] perturbations

An approximate solution to the angles-only initial relative orbit determination problem is developed and evaluated in the context of the two-body problem with J2 perturbations. The algorithm is non-iterative and requires the singular value decomposition of a 6 × 6 matrix and the solution of a fifteenth-order polynomial. The performance of the algorithm is investigated for non-circular low-Earth orbits and near-circular geostationary orbits with and without measurement error.

Bertrand’s theorem and virial theorem in fractional classical mechanics

Fractional classical mechanics is the classical counterpart of fractional quantum mechanics. The central force problem in this theory is investigated. Bertrand's theorem is generalized, and virial theorem is revisited, both in three spatial dimensions. In order to produce stable, closed, non-circular orbits, the inverse-square law and the Hooke's law should be modified in fractional classical mechanics.

First law of compact binary mechanics with gravitational-wave tails

We derive the first law of binary point-particle mechanics for generic bound (i.e. eccentric) orbits at the fourth post-Newtonian (4PN) order, accounting for the non-locality in time of the dynamics due to the occurence of a gravitational-wave tail effect at that order. Using this first law, we show how the periastron advance of the binary system can be related to the averaged redshift of one of the two bodies for a slightly non-circular orbit, in the limit where the eccentricity vanishes. Combining this expression with existing analytical self-force results for the averaged redshift, we recover the known 4PN expression for the circular-orbit periastron advance, to linear order in the mass ratio.

Task-Driven Orbit Design and Implementation on a Robotic C-Arm System for Cone-Beam CT.

This work applies task-driven optimization to the design of non-circular orbits that maximize imaging performance for a particular imaging task. First implementation of task-driven imaging on a clinical robotic C-arm system is demonstrated, and a framework for orbit calculation is described and evaluated. We implemented a task-driven imaging framework to optimize orbit parameters that maximize detectability index d'. This framework utilizes a specified Fourier domain task function and an analytical model for system spatial resolution and noise. Two experiments were conducted to test the framework. First, a simple task was considered consisting of frequencies lying entirely on the fz-axis (e.g., discrimination of structures oriented parallel to the central axial plane), and a "circle + arc" orbit was incorporated into the framework as a means to improve sampling of these frequencies, and thereby increase task-based detectability. The orbit was implemented on a robotic C-arm (Artis Zeego, Siemens Healthcare). A second task considered visualization of a cochlear implant simulated within a head phantom, with spatial frequency response emphasizing high-frequency content in the (fy , fz ) plane of the cochlea. An optimal orbit was computed using the task-driven framework, and the resulting image was compared to that for a circular orbit. For the fz -axis task, the circle + arc orbit was shown to increase d' by a factor of 1.20, with an improvement of 0.71 mm in a 3D edge-spread measurement for edges located far from the central plane and a decrease in streak artifacts compared to a circular orbit. For the cochlear implant task, the resulting orbit favored complementary views of high tilt angles in a 360° orbit, and d' was increased by a factor of 1.83. This work shows that a prospective definition of imaging task can be used to optimize source-detector orbit and improve imaging performance. The method was implemented for execution of non-circular, task-driven orbits on a clinical robotic C-arm system. The framework is sufficiently general to include both acquisition parameters (e.g., orbit, kV, and mA selection) and reconstruction parameters (e.g., a spatially varying regularizer).

Dinosaur Dynamics in an Integrative Model for Rotating Rocks and Deformable Bodies

Present long range interactions for signals, as well as large scale rare phenomena at Earthquakes are subjects of wave dynamics with large impact. Here, that will be addressed in a general (science fiction) format. The method used is based on energy balance for a continuum and kinematics of noncircular orbits, nco. Released vortex formation in a process invoking escape velocity for beamed up Dinosaurs, is modeled with fluid dynamics of a Helix. It is found that elasticity in terms of volumetric changes and vorticities interact giving large velocities and interchange of potential and kinetic energies. The model is an example of atmosphere integrative physiology.

Completion of metric reconstruction for a particle orbiting a Kerr black hole

Vacuum perturbations of the Kerr metric can be reconstructed from the corresponding perturbation in either of the two Weyl scalars $\psi_0$ or $\psi_4$, using a procedure described by Chrzanowski and others in the 1970s. More recent work, motivated within the context of self-force physics, extends the procedure to metric perturbations sourced by a particle in a bound geodesic orbit. However, the existing procedure leaves undetermined a certain stationary, axially-symmetric piece of the metric perturbation. In the vacuum region away from the particle, this "completion" piece corresponds simply to mass and angular-momentum perturbations of the Kerr background, with amplitudes that are, however, a priori unknown. Here we present and implement a rigorous method for finding the completion piece. The key idea is to impose continuity, off the particle, of certain gauge-invariant fields constructed from the full (completed) perturbation, in order to determine the unknown amplitude parameters of the completion piece. We implement this method in full for bound (eccentric) geodesic orbits in the equatorial plane of the Kerr black hole. Our results provide a rigorous underpinning of recent results by Friedman {\it et al.}\ for circular orbits, and extend them to non-circular orbits.

Metrics in the space of orbits and their application to searching for celestial objects of common origin

Finding a common origin for various celestial bodies, especially the relations between meteoroid streams, comets and asteroids (possibly extinct comets), remains one of the important problems in Solar system astronomy. Different criteria, starting with one by Southworth and Hawkins, have been used for this purpose. Ideally, they must represent some kind of metric in a five-dimensional space of orbits. Unfortunately, they are not ideal. The majority of the criteria have been examined by us. It turns out that they all represent pseudometrics for which the triangle axiom is not fulfilled. Besides this, they are inapplicable if at least one of the orbits is circular. We propose metrics free of all the aforementioned drawbacks. In addition, the metric properties of three factor-spaces, where orbits are identified irrespective of the values of node longitudes, pericentre arguments or both, are examined. The results are applied to the problem of searching for minor bodies of the Solar system with a common origin. The relationship between comet 96P/Machholz 1 and asteroid 2003EH1, as well as that between comet 2P/Encke and asteroid 2004TG10, has been proved. Using all criteria considered and the new metrics leads to practically identical results. This is explained by the fact that only close and essentially non-circular orbits were analysed. Besides this, the measure of orbit triples for which the triangle axiom failed is likely small, though this problem has not been examined yet.

Stability of the celestial body orbits under the influence of Yukawa potential

The famous Yukawa-type potential was very often used for explaining a great variety of recent observed astronomical phenomena which range from the solar-system scale to cosmological distances. In this paper we tackle the two-body problem in the Yukawa post-Newtonian field from the particular standpoint of orbits stability. Starting from the equations of motion and first integrals written in standard polar coordinates, we apply McGehee-type transformations of the second kind. Then we depict the phase-space structure considering the foliations by the energy constant and the angular momentum constant. Various stability regions are found for each case. The problem presents interesting features, such as: cases when all trajectories (maybe except for a separatrix) are stable; cases when there exist totally different types of orbits, both stable and unstable, for the same values of the energy constant and the angular momentum constant; the existence of stable motion for nonnegative energy levels; positive Lebesgue measure for initial data leading to quasiperiodic and noncircular periodic orbits; the key role of the angular momentum.

WE‐AB‐BRA‐08: Correction of Patient Motion in C‐Arm Cone‐Beam CT Using 3D‐2D Registration

Purpose:Intraoperative C‐arm cone‐beam CT (CBCT) is subject to artifacts arising from patient motion during the fairly long (∼5–20 s) scan times. We present a fiducial free method to mitigate motion artifacts using 3D‐2D image registration that simultaneously corrects residual errors in geometric calibration.Methods:A 3D‐2D registration process was used to register each projection to DRRs computed from the 3D image by maximizing gradient orientation (GO) using the CMA‐ES optimizer. The resulting rigid 6 DOF transforms were applied to the system projection matrices, and a 3D image was reconstructed via model‐based image reconstruction (MBIR, which accommodates the resulting noncircular orbit). Experiments were conducted using a Zeego robotic C‐arm (20 s, 200°, 496 projections) to image a head phantom undergoing various types of motion: 1) 5° lateral motion; 2) 15° lateral motion; and 3) 5° lateral motion with 10 mm periodic inferior‐superior motion. Images were reconstructed using a penalized likelihood (PL) objective function, and structural similarity (SSIM) was measured for axial slices of the reconstructed images. A motion‐free image was acquired using the same protocol for comparison.Results:There was significant improvement (p < 0.001) in the SSIM of the motion‐corrected (MC) images compared to uncorrected images. The SSIM in MC‐PL images was >0.99, indicating near identity to the motion‐free reference. The point spread function (PSF) measured from a wire in the phantom was restored to that of the reference in each case.Conclusion:The 3D‐2D registration method provides a robust framework for mitigation of motion artifacts and is expected to hold for applications in the head, pelvis, and extremities with reasonably constrained operative setup. Further improvement can be achieved by incorporating multiple rigid components and non‐rigid deformation within the framework. The method is highly parallelizable and could in principle be run with every acquisition.Research supported by National Institutes of Health Grant No. R01‐EB‐017226 and academic‐industry partnership with Siemens Healthcare (AX Division, Forcheim, Germany).

Self-calibration of cone-beam CT geometry using 3D–2D image registration

Robotic C-arms are capable of complex orbits that can increase field of view, reduce artifacts, improve image quality, and/or reduce dose; however, it can be challenging to obtain accurate, reproducible geometric calibration required for image reconstruction for such complex orbits. This work presents a method for geometric calibration for an arbitrary source-detector orbit by registering 2D projection data to a previously acquired 3D image. It also yields a method by which calibration of simple circular orbits can be improved. The registration uses a normalized gradient information similarity metric and the covariance matrix adaptation-evolution strategy optimizer for robustness against local minima and changes in image content. The resulting transformation provides a ‘self-calibration’ of system geometry. The algorithm was tested in phantom studies using both a cone-beam CT (CBCT) test-bench and a robotic C-arm (Artis Zeego, Siemens Healthcare) for circular and non-circular orbits. Self-calibration performance was evaluated in terms of the full-width at half-maximum (FWHM) of the point spread function in CBCT reconstructions, the reprojection error (RPE) of steel ball bearings placed on each phantom, and the overall quality and presence of artifacts in CBCT images. In all cases, self-calibration improved the FWHM—e.g. on the CBCT bench, FWHM = 0.86 mm for conventional calibration compared to 0.65 mm for self-calibration (p < 0.001). Similar improvements were measured in RPE—e.g. on the robotic C-arm, RPE = 0.73 mm for conventional calibration compared to 0.55 mm for self-calibration (p < 0.001). Visible improvement was evident in CBCT reconstructions using self-calibration, particularly about high-contrast, high-frequency objects (e.g. temporal bone air cells and a surgical needle). The results indicate that self-calibration can improve even upon systems with presumably accurate geometric calibration and is applicable to situations where conventional calibration is not feasible, such as complex non-circular CBCT orbits and systems with irreproducible source-detector trajectory.

The Search for Extraterrestrial Intelligence in Earth's Solar Transit Zone.

Over the past few years, astronomers have detected thousands of planets and candidate planets by observing their periodic transits in front of their host stars. A related method, called transit spectroscopy, might soon allow studies of the chemical imprints of life in extrasolar planetary atmospheres. Here, we address the reciprocal question, namely, from where is Earth detectable by extrasolar observers using similar methods. We explore Earth's transit zone (ETZ), the projection of a band around Earth's ecliptic onto the celestial plane, where observers can detect Earth transits across the Sun. ETZ is between 0.520° and 0.537° wide due to the noncircular Earth orbit. The restricted Earth transit zone (rETZ), where Earth transits the Sun less than 0.5 solar radii from its center, is about 0.262° wide. We first compile a target list of 45 K and 37 G dwarf stars inside the rETZ and within 1 kpc (about 3260 light-years) using the Hipparcos catalogue. We then greatly enlarge the number of potential targets by constructing an analytic galactic disk model and find that about 10(5) K and G dwarf stars should reside within the rETZ. The ongoing Gaia space mission can potentially discover all G dwarfs among them (several 10(4)) within the next 5 years. Many more potentially habitable planets orbit dim, unknown M stars in ETZ and other stars that traversed ETZ thousands of years ago. If any of these planets host intelligent observers, they could have identified Earth as a habitable, or even as a living, world long ago, and we could be receiving their broadcasts today. The K2 mission, the Allen Telescope Array, the upcoming Square Kilometer Array, or the Green Bank Telescope might detect such deliberate extraterrestrial messages. Ultimately, ETZ would be an ideal region to be monitored by the Breakthrough Listen Initiatives, an upcoming survey that will constitute the most comprehensive search for extraterrestrial intelligence so far.

Feeding and feedback in NGC 3081

We present two-dimensional gaseous kinematics of the inner 1.2 $\times$ 1.8 kpc$^2$ of the Seyfert 2 galaxy NGC3081, from optical spectra (5600--7000\r{A}) obtained with the GMOS integral field spectrograph on the Gemini North telescope at a spatial resolution of $\approx$ 100pc. We have identified two-components in the line emitting gas. A narrower component (FWHM $\approx$ 60-100km s$^{-1}$), which appears to be gas in the galaxy disk, and which shows a distorted rotation pattern, is observed over the whole field of view. A broader component (FWHM $\approx$150-250 km s$^{-1}$) is present in the inner $\approx$ 2arcsec (200pc) and shows blueshifts and redshifts in the near and far sides of the galaxy, respectively, consistent with a bipolar outflow. Assuming this to be the case, we estimate that the mass outflow rate in ionized gas ($\dot{M}_{out}$) is between 1.9 $\times 10^{-3}$M$_{\odot}$ yr$^{-1}$ and 6.9 $\times 10^{-3}$M$_{\odot}$ yr$^{-1}$. The subtraction of a rotation model from the narrower component velocity field reveals a pattern of excess blueshifts of $\approx$ 50km s$^{-1}$ in the far side of the galaxy and similar excess redshifts in the near side, which are cospatial with a previously known nuclear bar. We interpret these residuals as due to gas following non-circular orbits in the barred potential. Under the assumption that these motions may lead to gas inflows, we estimate an upper limit for the mass inflow rate in ionized gas of $\phi$ $\approx$ 1.3 $\times 10^{-2}$M$_{\odot}$ yr$^{-1}$.

Near-horizon circular orbits and extremal limit for dirty rotating black holes

We consider generic rotating axially symmetric "dirty" (surrounded by matter) black holes. Near-horizon circular equatorial orbits are examined in two different cases of near-extremal (small surface gravity $\kappa $) and exactly extremal black holes. This has a number of qualitative distinctions. In the first case, it is shown that such orbits can lie as close to the horizon as one wishes on suitably chosen slices of space-time when $\kappa \rightarrow 0$. This generalizes observation of T.\ Jacobson Class. Quantum Grav. 28 187001 (2011) made for the Kerr metric. If a black hole is extremal ($\kappa =0$), circular on-horizon orbits are impossible for massive particles but, in general, are possible in its vicinity. The corresponding black hole parameters determine also the rate with which a fine-tuned particle on the noncircular near-horizon orbit asymptotically approaches the horizon. Properties of orbits under discussion are also related to the Ba% \~nados-Silk-West effect of high energy collisions near black holes. Impossibility of the on-horizon orbits in question is manifestation of kinematic censorship that forbids infinite energies in collisions.

Comparison between self-force and post-Newtonian dynamics: Beyond circular orbits

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modelling the dynamics of compact binary systems. Comparison of their results in an overlapping domain of validity provides a crucial test for both methods, and can be used to enhance their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first time, we extend such comparisons to noncircular orbits---specifically, to a system of two nonspinning objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged quantity $\langle U \rangle$ that generalizes Detweiler's redshift invariant. The functional relationship $\langle U \rangle(\Omega_r,\Omega_\phi)$, where $\Omega_r$ and $\Omega_\phi$ are the frequencies of the radial and azimuthal motions, is an invariant characteristic of the conservative dynamics. We compute $\langle U \rangle(\Omega_r,\Omega_\phi)$ numerically through linear order in the mass ratio $q$, using a GSF code which is based on a frequency-domain treatment of the linearized Einstein equations in the Lorenz gauge. We also derive $\langle U \rangle(\Omega_r,\Omega_\phi)$ analytically through 3PN order, for an arbitrary $q$, using the known near-zone 3PN metric and the generalized quasi-Keplerian representation of the motion. We demonstrate that the $\mathcal{O}(q)$ piece of the analytical PN prediction is perfectly consistent with the numerical GSF results, and we use the latter to estimate yet unknown pieces of the 4PN expression at $\mathcal{O}(q)$.

On feathers, bifurcations and shells: the dynamics of tidal streams across the mass scale

I present an organic description of the regimes of collisionless tidal streams and define the orderings between the physical quantities that shape their morphology. Three fundamental dichotomies are identified in the form of dimensionless inequalities. These govern i) the speed of the stream's growth, ii) its internal coherence, iii) its thickness or opening angles. The mechanisms that regulate such main properties are analysed. The slope of the host's density profile influences the speed of the stream's growth, in both length and width, as steeper profiles enhance differential streaming. Internal coherence is the requirement for the appearance of substructure in tidal debris, and I concentrate on the `feathering' typical of GC streams. Overdensities are associated with minima in the relative streaming velocity of the stream members. For streams with high circularity, these are caused by the epicyclic oscillations of stars; however, for highly non-circular progenitor's orbits, substructure is caused by the oscillating differences in energy and actions with which material is shed at different orbital phases of the progenitor. This modulation results in different streaming speeds: the streakline of material shed between two successive apocentric passages is folded along its length, pulled at its centre by the faster streaming of particles released near pericenter, which are therefore more widely scattered. When the stream is coherent enough, this mechanism is potentially capable of generating a bimodal profile in the density distributions of the longer wraps of more massive progenitors, which I dub `bifurcations'. The conditions for internal coherence are explored and I comment on the cases of Palomar 5, Willman 1, the Anticenter and Sagittarius' streams. Analytical methods are accompanied by numerical experiments, performed using a purposely built generative model, also presented here.

Self-Calibration of Cone-Beam CT Geometry Using 3D-2D Image Registration: Development and Application to Task-Based Imaging with a Robotic C-Arm.

Robotic C-arm systems are capable of general noncircular orbits whose trajectories can be driven by the particular imaging task. However obtaining accurate calibrations for reconstruction in such geometries can be a challenging problem. This work proposes a method to perform a unique geometric calibration of an arbitrary C-arm orbit by registering 2D projections to a previously acquired 3D image to determine the transformation parameters representing the system geometry. Experiments involved a cone-beam CT (CBCT) bench system, a robotic C-arm, and three phantoms. A robust 3D-2D registration process was used to compute the 9 degree of freedom (DOF) transformation between each projection and an existing 3D image by maximizing normalized gradient information with a digitally reconstructed radiograph (DRR) of the 3D volume. The quality of the resulting "self-calibration" was evaluated in terms of the agreement with an established calibration method using a BB phantom as well as image quality in the resulting CBCT reconstruction. The self-calibration yielded CBCT images without significant difference in spatial resolution from the standard ("true") calibration methods (p-value >0.05 for all three phantoms), and the differences between CBCT images reconstructed using the "self" and "true" calibration methods were on the order of 10-3 mm-1. Maximum error in magnification was 3.2%, and back-projection ray placement was within 0.5 mm. The proposed geometric "self" calibration provides a means for 3D imaging on general non-circular orbits in CBCT systems for which a geometric calibration is either not available or not reproducible. The method forms the basis of advanced "task-based" 3D imaging methods now in development for robotic C-arms.

-Infinity Leveraging Boundary-Value Problem and Application in Spacecraft Trajectory Design

Interplanetary and intermoon tour missions have benefited from the implementation of leveraging maneuvers that efficiently change spacecraft energy relative to a flyby body. In the current work, these leveraging maneuvers are generalized and reformulated into a boundary-value problem suitable for broad trajectory searches using only one additional continuous degree of freedom. The presented method naturally incorporates noncircular orbits and arbitrary ephemeris locations for bodies as well as leveraging maneuvers between different bodies. These advantages come from using the same boundary conditions as the Lambert problem, which also maintains the Lambert-based outer-loop search architecture used in many trajectory design tools. The general maneuver formulation of the presented method allows it to be used for interplanetary or intermoon trajectories and allows the incorporation of nontangent leveraging transfers. Some examples of the new formulation’s utility are also explored using representative transfers; these examples present some of the families of leveraging solutions and show that bielliptic transfers are a special case of interbody leveraging.

A new analysis of Galileo dust data near Jupiter

The Galileo Dust Detection System (DDS) detected a population of micron-sized grains in and amongst the orbits of Io, Europa, Ganymede and Callisto. Previous studies, using roughly 50% of the data now available, concluded that the dominant sources for the impacts were magnetospherically captured interplanetary particles largely on retrograde orbits (Colwell et al., 1998b; Thiessenhusen et al., 2000) and impact-generated ejecta from the Galilean satellites (Krüger et al., 1999b; Krivov et al., 2002a). Here we revisit the problem with the full data set and broaden our consideration to include four additional source populations: debris from the outer satellites, interplanetary and interstellar grains and particles accelerated outwards from Io and the jovian rings. We develop a model of detectable orbits at each Galileo position and we find that about 10% of the impact data require non-circular orbits with eccentricities greater than 0.1. In addition, ~3% of impacts require orbital solutions with eccentricities in excess of 0.7. Using the spatial distribution of particles, we are able to exclude, as dominant sources, all the additional source populations except for outer satellite particles. A study of DDS directional information demonstrates that none of the six standard sources fit the data well and thus a combination of sources is necessary. There are insufficient data to uniquely identify the relative strengths of the various contributions. However, we find an excess of large particles that is consistent with retrograde trajectories.