386 research outputs found
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
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