Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.     

Statistical Models for the Solar Flare Interval Distribution in Individual Active Regions

This article discusses statistical models for the solar flare interval distribution in individual active regions. We analyzed solar flare data in 55 active regions that are listed in the Geosynchronous Operational Environmental Satellite (GOES) soft X-ray flare catalog for the years from 1981 to 2005. We discuss some problems with a conventional procedure to derive probability density functions from any data set and propose a new procedure, which uses the maximum likelihood method and Akaike Information Criterion (AIC) to objectively compare some competing probability density functions. Previous studies of the solar flare interval distribution in individual active regions only dealt with constant or time-dependent Poisson process models, and no other models were discussed. We examine three models – exponential, lognormal, and inverse Gaussian – as competing models for probability density functions in this study. We found that lognormal and inverse Gaussian models are more likely models than the exponential model for the solar flare interval distribution in individual active regions. The possible solar flare mechanisms for the distribution models are briefly mentioned. We also briefly investigated the time dependence of probability density functions of the solar flare interval distribution and found that some active regions show time dependence for lognormal and inverse Gaussian distribution functions. The results suggest that solar flares do not occur randomly in time; rather, solar flare intervals appear to be regulated by solar flare mechanisms. Determining a solar flare interval distribution is an essential step in probabilistic solar flare forecasting methods in space weather research. We briefly mention a probabilistic solar flare forecasting method as an application of a solar flare interval distribution analysis. The application of our distribution analysis to a probabilistic solar flare forecasting method is one of the main objectives of this study.     

The 6C** sample of steep-spectrum radio sources – II. Redshift distribution and the space density of high-redshift radio galaxies

We use the 6C** sample to investigate the comoving space density of powerful, steep-spectrum radio sources. This sample, consisting of 68 objects, has virtually complete K -band photometry and spectroscopic redshifts for 32 per cent of the sources. In order to find its complete redshift distribution, we develop a method of redshift estimation based on the K – z diagram of the 3CRR, 6CE, 6C* and 7CRS radio galaxies. Based on this method, we derive redshift probability density functions for all the optically identified sources in the 6C** sample. Using a combination of spectroscopic and estimated redshifts, we select the most radio luminous sources in the sample. Their redshift distribution is then compared with the predictions of the radio luminosity function of Jarvis et al. We find that, within the uncertainties associated with the estimation method, the data are consistent with a constant comoving space density of steep-spectrum radio sources beyond z ≳ 2.5, and rule out a steep decline.     

Semi-analytic approach to understanding the distribution of neutral hydrogen in the Universe

Analytic derivations of the correlation function and the column density distribution for neutral hydrogen in the intergalactic medium (IGM) are presented, assuming that the non-linear baryonic mass density distribution in the IGM is lognormal. This ansatz was used earlier by Bi & Davidsen to perform one-dimensional simulations of lines of sight and analyse the properties of absorption systems. We have taken a completely analytic approach, which allows us to explore a wide region of the parameter space for our model. The analytic results have been compared with observations to constrain various cosmological and IGM parameters, whenever possible. Two kinds of correlation functions are defined: (i) along the line of sight (LOS); and (ii) across the transverse direction. We find that the effects on the LOS correlation owing to changes in cosmology and the slope of the equation of state of the IGM, γ , are of the same order, which means that we cannot constrain both the parameters simultaneously. However, it is possible to constrain γ and its evolution using the observed LOS correlation function at different epochs provided that one knows the background cosmology. We suggest that the constraints on the evolution of γ obtained using the LOS correlation can be used as an independent tool to probe the reionization history of the Universe. From the transverse correlation function, we obtain the excess probability, over random, of finding two neutral hydrogen overdense regions separated by an angle θ . We find that this excess probability is always less than 1 per cent for redshifts greater than 2. Our models also reproduce the observed column density distribution for neutral hydrogen, and the shape of the distribution depends on γ . Our calculations suggest that one can rule out γ >1.6 for z ≃2.31 using the column density distribution. However, one cannot rule out higher values of γ at higher redshifts.     

Asteroid orbits using phase-space volumes of variation

We present a statistical orbit computation technique for asteroids with transitional observational data, that is, a moderate number of data points spanning a moderate observational time interval. With the help of local least-squares solutions in the phase space of the orbital elements, we map the volume of variation as a function of one or more of the elements. We sample the resulting volume using a Monte Carlo technique and, with proper weights for the sample orbital elements, characterize the six-dimensional orbital-element probability density function. The volume-of-variation (VOV) technique complements the statistical ranging technique for asteroids with exiguous observational data (short time intervals and/or small numbers of observations) and the least-squares technique for extensive observational data. We show that, asymptotically, results using the new technique agree closely with those from ranging and least squares. We apply the technique to the near-Earth object 2004 HA₃₉, the main-belt object 2004 QR and the transneptunian object 2002 CX₂₂₄ recently observed at the Nordic Optical Telescope on La Palma, illustrating the potential of the technique in ephemeris prediction. The VOV technique helps us assess the phase transition in orbital-element probability densities, that is, the non-linear collapse of wide orbital-element distributions to narrow localized ones. For the three objects above, the transition takes place for observational time intervals of the order of 10 h, 5 d and 10 months, respectively, emphasizing the significance of the orbital-arc fraction covered by the observations.     

Estimation of Distribution Algorithm for Initial Orbit Determination of Too-short-arc Based on Kernel Density Estimation

A new approach of the initial orbit determination for too-short-arc with angular measurements is implemented by building the probabilistic model in the solution space with the estimation of distribution algorithm. Without any assumption about distribution, the non-parametric kernel density estimation is employed in the model building. The method, unlike other evolutionary algorithms, such as the genetic algorithm and particle swarm optimization, considers the fitness as well as the characteristics of the solution space. Numerical experiments with real observations indicate that without any constraints, the proposed technique has a good performance for the observations of usual accuracy.     

Background–source separation in astronomical images with Bayesian probability theory – I. The method

A probabilistic technique for the joint estimation of background and sources with the aim of detecting faint and extended celestial objects is described. Bayesian probability theory is applied to gain insight into the co-existence of background and sources through a probabilistic two-component mixture model, which provides consistent uncertainties of background and sources. A multiresolution analysis is used for revealing faint and extended objects in the frame of the Bayesian mixture model. All the revealed sources are parametrized automatically providing source position, net counts, morphological parameters and their errors.
We demonstrate the capability of our method by applying it to three simulated data sets characterized by different background and source intensities. The results of employing two different prior knowledge on the source signal distribution are shown. The probabilistic method allows for the detection of bright and faint sources independently of their morphology and the kind of background. The results from our analysis of the three simulated data sets are compared with other source detection methods. Additionally, the technique is applied to ROSAT All-Sky Survey data.     

Venus gravity west of Beta Regio

Doppler tracking data from the Pioneer Venus Orbiter (PVO) have been used to estimate the anomalous gravity field in the region of Venus west of Beta Regio. The analysis invokes a Kalman filter-smoother to solve the nonlinear spacecraft state estimation problem and a linear Bayesian estimator to perform the geophysical inversion. The topographic map for this region, derived from the PVO radar, has been filtered to have the same distortions and degree of smoothing as the gravity map. The undulations of the gravity are about 0.2 times as large as expected from the topography on the assumption that the latter is uncompensated. A comparison of the gravity and topography by means of the spectral admittance is consistent with Airy compensation at a depth of 50 km if the surface material has a density of 2.6 g/cm³. However, this is not a unique interpretation.     

The entropic prior for distributions with positive and negative values

The maximum entropy method has been used to reconstruct images in a wide range of astronomical fields, but in its traditional form it is restricted to the reconstruction of strictly positive distributions. We present an extension of the standard method to include distributions that can take both positive and negative values. The method may therefore be applied to a much wider range of astronomical reconstruction problems. In particular, we derive the form of the entropy for positive/negative distributions and use direct counting arguments to find the form of the entropic prior. We also derive the measure on the space of positive/negative distributions, which allows the definition of probability integrals and hence the proper quantification of errors.     

The distance–redshift equation in quintessence cosmology and the estimation of H0 through time delays

We calculate analytically and numerically the distance–redshift equation in perfect fluid quintessence models and give an accurate fit to the numerical solutions for all the values of the density parameter and the quintessence equation of state. Then we apply our solutions to the estimation of H ₀ from multiple image time delays and find that the inclusion of quintessence modifies significantly the likelihood distribution of H ₀, generally reducing the best estimate with respect to a pure cosmological constant. Marginalizing over the other parameters (Ω _m and the quintessence equation of state) we obtain H ₀=71±6 km s⁻¹ Mpc⁻¹ for an empty beam and H ₀=64±4 km s⁻¹ Mpc⁻¹ for a filled beam. These errors, however, do not take into account the uncertainty on the modelling of the lens. We also discuss the future prospects for distinguishing quintessence from a cosmological constant with time delays.     

Analytical solutions for the equations of motion of a space vehicle during the atmospheric re-entry phase on a 2-D trajectory

A practical and important problem encountered during the atmospheric re-entry phase is to determine analytical solutions for the space vehicle dynamical equations of motion. The author proposes new solutions for the equations of trajectory and flight-path angle of the space vehicle during the re-entry phase in Earth’s atmosphere. Explicit analytical solutions for the aerodynamic equations of motion can be effectively applied to investigate and control the rocket flight characteristics. Setting the initial conditions for the speed, re-entering flight-path angle, altitude, atmosphere density, lift and drag coefficients, the nonlinear differential equations of motion are linearized by a proper choice of the re-entry range angles. After integration, the solutions are expressed with the Exponential Integral, and Generalized Exponential Integral functions. Theoretical frameworks for proposed solutions as well as, several numerical examples, are presented.     

Echo mapping of Swift J1753.5−0127

Caustics are a generic feature of the non-linear growth of structure in the dark matter distribution. If the dark matter were absolutely cold, its mass density would diverge at caustics, and the integrated annihilation probability would also diverge for individual particles participating in them. For realistic dark matter candidates, this behaviour is regularized by small but non-zero initial thermal velocities. We present a mathematical treatment of evolution from hot, warm or cold dark matter initial conditions which can be directly implemented in cosmological N -body codes. It allows the identification of caustics and the estimation of their annihilation radiation in fully general simulations of structure formation.     

Space Event and Outlier Detection Based on Expectation Maximization Algorithm

The United States provide Element Sets (ELSET) database in Two-Line Element (TLE) format for public use, which plays an important role in the inversion of atmospheric density in the thermosphere, ballistic coefficient estimation, early-warning and so on. Due to large uncertainties existing in the TLE generation process, space environment changes, and space events, ELSET database contains a large number of abnormal TLE data to be filtered, such as corrected TLE, orbital element outlier, and Bstar outlier. The existing methods to filter out the outliers lack general applicability and are very complicated, which are only applicable to a few space targets in certain orbit regions. To overcome the shortcomings of the existing methods, a filtering method is proposed based on Expectation Maximization (EM) algorithm employing a sliding window and polynomial fitting method, which can detect outliers for different orbital elements and space events. The research shows that the algorithm can effectively single out the outliers in TLE sequences and is suitable for all orbital debris.     

About averaging procedures in the problem of asteroid motion

A new mathematically correct approach to construct an averaging procedure for the motion of a massless body around the central body perturbed by fully interacting planets is developed and the errors of the standard solution are discussed. The new technique allows to combine the advantages of the Hamiltonian representation with the usage of standard osculating elements in combination with all the standard expansions of the perturbing functions. The main idea is to introduce new additional variables conjugate to all the standard elements and to work in a corresponding super phase space. In this way, the number of variables is doubled at first, but one has to deal with only one Hamiltonian. The artificially introduced variables disappear from the final averaged equations as well as from the transformation formulae connecting the osculating and the mean elements.     

Statistical biases and errors inherent in photoclinometric surface slope estimation with natural light

Photoclinometry is the most common method used to obtain high-resolution topographic maps of planetary terrain. We derive the likelihood function of photoclinometric surface slope from (1) the probability distribution of the measured photon count of natural sunlight through a Charge-Coupled Device (CCD) including uncertainty due to camera shot noise, camera read noise, small-scale albedo fluctuation and atmospheric haze, and (2) a photometric model relating photocount to surface orientation. We then use classical estimation theory to determine the theoretically exact biases and errors inherent in photoclinometric surface slope and show when they may be approximated by asymptotic expressions for sufficiently high sample size. We show how small-scale albedo variability often dominates biases and errors, which may become an order of magnitude larger than surface slopes when surface reflectance has a weak dependence on surface tilt. We provide bounds on the minimum possible error of any unbiased photoclinometric surface slope estimate, and compute the sample sizes necessary to constrain errors within desired design thresholds.     

Transformations of the perturbed two-body problem to unperturbed harmonic oscillators

Transformations are given which change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients. The orginally singular, nonlinear and Lyapunov unstable equations become in this way regular, linear, and the stable solution may be written down immediately in terms of the new variables. Transformations of the independent and dependent variables are treated separately as well as jointly. Using arbitrary and special functions for the transformations allows a systematic discussion of previously introduced and new anomalies. For the unperturbed two-body problem the theorem is proved according to which if the transformations are power-functions of the radial variable, then only the eccentric and the true anomalies with the corresponding transformations of the radial variable will result in harmonic oscillators. Important practical applications are to increase autonomous operations in space, since by replacing lengthy numerical integrations by transformations, computer requirements are significantly reduced.     

Asteroid identification over apparitions

We present a new method for the linking of scarce asteroid astrometry over apparitions, and apply it both to simulated and real data to prove its feasibility. Up to date, there has not been a robust method available to search for linkages between the approximately 50,000 provisionally designated sets of asteroid astrometry spanning less than two days. Unless such a scarce set of astrometry is linked to another set of astrometry, the underlying object can be considered lost as the ephemeris uncertainties are substantial. The new method, which can tackle the challenges, is based on Ranging, which is a fully nonlinear, statistical orbital inversion method. Ranging properly treats astrometric uncertainties and propagates the uncertainty to the resulting orbital-element probability density, which is sampled by a set of orbits. The new orbital-element-space multiple-address-comparison (oMAC) method uses dimensionality-reduction techniques and tree structures to efficiently search for overlapping probability densities in the orbital-element phase space. Overlapping probability densities indicate a candidate linkage between astrometric observation sets. To accept a candidate linkage, we have to find a many-body orbital solution which reproduces the observed positions within the observational uncertainties. To find the linking orbit, we use a multi-step approach starting from a Monte-Carlo generation of possible orbits in a reduced volume of the orbital-element phase space and ending with a least-squares orbital solution, which, in addition to the Sun's gravitation, also takes into account the gravitational influence of the relevant planets. The new multiple-address-comparison method has a loglinear computational complexity, that is, it scales as O(nlogn), where n is the number of included observation sets. It has recently also been implemented for the ephemeris-space multiple-address-comparison (eMAC) method, which is optimized for the short-term linking of scarce astrometry.     

泊松分布下的非线性拟合

对一数据涨落服从泊松概率分布的实验数据组进行拟合,如果拟合函数是非线性的,常常只能采用某些近似方法。近似性之一,表现在将拟合函数线性化,或者采用某种参数寻优的方法。近似性之二,则是将泊松问题近似地看作高斯分布问题。本文从最大或然法出发,对两类不同概率分布情况下的曲线拟合进行了比较,给出了对泊松分布问题更严格的求解途径。对于特殊的非线性拟合函数y(x)=ae~(bx~N),直接给出了迭代解。本文对曲线拟合的“面积亏损”问题,也进行了讨论。     

Time evolution of orbital uncertainties for the impactor candidate 2004 AS1

We evaluate the asteroid impact risk from the discovery night onwards using six-dimensional statistical orbit computation techniques to examine the a posteriori probability density of the orbital elements. Close to the discovery moment the observational data of an object are typically exiguous: the number of observations is very small and/or the covered orbital arc is very short. For such data, the covariance matrices computed in the linear approximation (e.g., with the least-squares technique) are known to fail to describe the uncertainties in the orbital parameters. The technique of statistical ranging gives us rigorous means to assess the orbital uncertainties already on the discovery night. To examine the time evolution of orbital uncertainties, we make use of a new nonlinear Monte Carlo technique of phase-space sampling using volumes of variation, which complements the ranging technique for exiguous data and the least-squares technique for extensive observational data. We apply the statistical techniques to the near-Earth Asteroid 2004 AS₁, which grabbed the attention of asteroid scientists because, for one day, it posed the highest and most immediate impact risk so far recorded. We take this extreme case to illustrate the ambiguities in the impact risk assessment for short arcs. We confirm that the weighted fraction of the collision orbits at discovery was large but conclude that this was mostly due to the discordance of the discovery-night observations. This case study highlights the need to introduce a regularization in terms of an a priori probability density to secure the invariance of the probabilistic analysis especially in the nonlinear orbital inversion for short arcs. We remark that a predominant role of the a priori can give indications of the feasibility of the probabilistic interpretation, that is, how reliable the results derived from the a posteriori probability density are. Nevertheless, the strict mathematical definition of, e.g., the collision probability remains valid, and our nonlinear statistical techniques give us the means to always deduce, at the very least, order-of-magnitude-estimates for the collision probability.