Performance analysis of the Least-Squares estimator in Astrometry

We characterize the performance of the widely-used least-squares estimator in astrometry in terms of a comparison with the Cramer-Rao lower variance bound. In this inference context the performance of the least-squares estimator does not offer a closed-form expression, but a new result is presented (Theorem 1) where both the bias and the mean-square-error of the least-squares estimator are bounded and approximated analytically, in the latter case in terms of a nominal value and an interval around it. From the predicted nominal value we analyze how efficient is the least-squares estimator in comparison with the minimum variance Cramer-Rao bound. Based on our results, we show that, for the high signal-to-noise ratio regime, the performance of the least-squares estimator is significantly poorer than the Cramer-Rao bound, and we characterize this gap analytically. On the positive side, we show that for the challenging low signal-to-noise regime (attributed to either a weak astronomical signal or a noise-dominated condition) the least-squares estimator is near optimal, as its performance asymptotically approaches the Cramer-Rao bound. However, we also demonstrate that, in general, there is no unbiased estimator for the astrometric position that can precisely reach the Cramer-Rao bound. We validate our theoretical analysis through simulated digital-detector observations under typical observing conditions. We show that the nominal value for the mean-square-error of the least-squares estimator (obtained from our theorem) can be used as a benchmark indicator of the expected statistical performance of the least-squares method under a wide range of conditions. Our results are valid for an idealized linear (one-dimensional) array detector where intra-pixel response changes are neglected, and where flat-fielding is achieved with very high accuracy.Comment: 35 pages, 8 figures. Accepted for publication by PAS

Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object

Context. The best precision that can be achieved to estimate the location of a stellar-like object is a topic of permanent interest in the astrometric community. Aims. We analyse bounds for the best position estimation of a stellar-like object on a CCD detector array in a Bayesian setting where the position is unknown, but where we have access to a prior distribution. In contrast to a parametric setting where we estimate a parameter from observations, the Bayesian approach estimates a random object (i.e., the position is a random variable) from observations that are statistically dependent on the position. Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimum mean square error (MMSE) of the best estimator of the position of a point source on a linear CCD-like detector, as a function of the properties of detector, the source, and the background. Results. We quantify and analyse the increase in astrometric performance from the use of a prior distribution of the object position, which is not available in the classical parametric setting. This gain is shown to be significant for various observational regimes, in particular in the case of faint objects or when the observations are taken under poor conditions. Furthermore, we present numerical evidence that the MMSE estimator of this problem tightly achieves the Bayesian CR bound. This is a remarkable result, demonstrating that all the performance gains presented in our analysis can be achieved with the MMSE estimator. Conclusions The Bayesian CR bound can be used as a benchmark indicator of the expected maximum positional precision of a set of astrometric measurements in which prior information can be incorporated. This bound can be achieved through the conditional mean estimator, in contrast to the parametric case where no unbiased estimator precisely reaches the CR bound.Comment: 17 pages, 12 figures. Accepted for publication on Astronomy & Astrophysic

Old Main-Sequence Turnoff Photometry in the Small Magellanic Cloud. I. Constraints on the Star Formation History in Different Fields

We present ground-based B and R-band color-magnitude diagrams (CMDs), reaching the oldest main-sequence turnoffs with good photometric accuracy for twelve fields in the Small Magellanic Cloud (SMC). Our fields, located between ~1 and ~4 degrees from the center of the galaxy, are situated in different parts of the SMC such as the "Wing'' area, and towards the West and South. In this paper we perform a first analysis of the stellar content in our SMC fields through comparison with theoretical isochrones and color functions (CFs). We find that the underlying spheroidally distributed population is composed of both intermediate-age and old stars and that its age composition does not show strong galacto-centric gradients. The three fields situated toward the east, in the Wing region, show very active current star formation. However, only in the eastern field closest to the center do we find an enhancement of recent star formation with respect to a constant SFR(t). The fields corresponding to the western side of the SMC present a much less populated young MS, and the CF analysis indicates that the SFR(t) greatly diminished around 2 Gyr ago in these parts. Field smc0057, the closest to the center of the galaxy and located in the southern part, shows recent star formation, while the rest of the southern fields present few bright MS stars. The structure of the red clump in all the CMDs is consistent with the large amount of intermediate-age stars inferred from the CMDs and color functions. None of the SMC fields presented here are dominated by old stellar populations, a fact that is in agreement with the lack of a conspicuous horizontal branch in all these SMC CMDs. This could indicate that a disk population is ruling over a possible old halo in all the observed fields.Comment: To appear in A

Orbits for eighteen visual binaries and two double-line spectroscopic binaries observed with HRCAM on the CTIO SOAR 4m telescope, using a new Bayesian orbit code based on Markov Chain Monte Carlo

We present orbital elements and mass sums for eighteen visual binary stars of spectral types B to K (five of which are new orbits) with periods ranging from 20 to more than 500 yr. For two double-line spectroscopic binaries with no previous orbits, the individual component masses, using combined astrometric and radial velocity data, have a formal uncertainty of ~0.1 MSun. Adopting published photometry, and trigonometric parallaxes, plus our own measurements, we place these objects on an H-R diagram, and discuss their evolutionary status. These objects are part of a survey to characterize the binary population of stars in the Southern Hemisphere, using the SOAR 4m telescope+HRCAM at CTIO. Orbital elements are computed using a newly developed Markov Chain Monte Carlo algorithm that delivers maximum likelihood estimates of the parameters, as well as posterior probability density functions that allow us to evaluate the uncertainty of our derived parameters in a robust way. For spectroscopic binaries, using our approach, it is possible to derive a self-consistent parallax for the system from the combined astrometric plus radial velocity data ("orbital parallax"), which compares well with the trigonometric parallaxes. We also present a mathematical formalism that allows a dimensionality reduction of the feature space from seven to three search parameters (or from ten to seven dimensions - including parallax - in the case of spectroscopic binaries with astrometric data), which makes it possible to explore a smaller number of parameters in each case, improving the computational efficiency of our Markov Chain Monte Carlo code.Comment: 32 pages, 9 figures, 6 tables. Detailed Appendix with methodology. Accepted by The Astronomical Journa

Optimality of the Maximum Likelihood estimator in Astrometry

The problem of astrometry is revisited from the perspective of analyzing the attainability of well-known performance limits (the Cramer-Rao bound) for the estimation of the relative position of light-emitting (usually point-like) sources on a CCD-like detector using commonly adopted estimators such as the weighted least squares and the maximum likelihood. Novel technical results are presented to determine the performance of an estimator that corresponds to the solution of an optimization problem in the context of astrometry. Using these results we are able to place stringent bounds on the bias and the variance of the estimators in close form as a function of the data. We confirm these results through comparisons to numerical simulations under a broad range of realistic observing conditions. The maximum likelihood and the weighted least square estimators are analyzed. We confirm the sub-optimality of the weighted least squares scheme from medium to high signal-to-noise found in an earlier study for the (unweighted) least squares method. We find that the maximum likelihood estimator achieves optimal performance limits across a wide range of relevant observational conditions. Furthermore, from our results, we provide concrete insights for adopting an adaptive weighted least square estimator that can be regarded as a computationally efficient alternative to the optimal maximum likelihood solution. We provide, for the first time, close-form analytical expressions that bound the bias and the variance of the weighted least square and maximum likelihood implicit estimators for astrometry using a Poisson-driven detector. These expressions can be used to formally assess the precision attainable by these estimators in comparison with the minimum variance bound.Comment: 24 pages, 7 figures, 2 tables, 3 appendices. Accepted by Astronomy & Astrophysic

Speckle interferometry at SOAR in 2019

The results of speckle interferometric observations at the 4.1 m Southern Astrophysical Research Telescope (SOAR) in 2019 are given, totaling 2555 measurements of 1972 resolved pairs with separations from 15 mas (median 0.21") and magnitude difference up to 6 mag, and non-resolutions of 684 targets. We resolved for the first time 90 new pairs or subsystems in known binaries. This work continues our long-term speckle program. Its main goal is to monitor orbital motion of close binaries, including members of high-order hierarchies and Hipparcos pairs in the solar neighborhood. We give a list of 127 orbits computed using our latest measurements. Their quality varies from excellent (25 orbits of grades 1 and 2) to provisional (47 orbits of grades 4 and 5).Comment: Accepted by The Astronomical Journal. 10 pages, 5 Figures. Measurements and non-resolutions, published electronically, are available from the first author. arXiv admin note: substantial text overlap with arXiv:1905.1043

Speckle interferometry at SOAR in 2015

The results of speckle interferometric observations at the SOAR telescope in 2015 are given, totalling 1303 measurements of 924 resolved binary and multiple stars and non-resolutions of 260 targets. The separations range from 12 mas to 3.37" (median 0.17"); the maximum measured magnitude difference is 6.7 mag. We resolved 27 pairs for the first time, including 10 as inner or outer subsystems in previously known binaries, e.g. the 50-mas pair in Epsilon Cha. Newly resolved pairs are commented upon. We discuss three apparently non-hierarchical systems discovered in this series, arguing that their unusual configuration results from projection. The resolved quadruple system HIP 71510 is studied as well.Comment: 10 pages, 8 figures. Accepted for publication in AJ. The online tables are not included, available from Tokovinin on request. arXiv admin note: text overlap with arXiv:1506.0571