Visual Binary Stars with Partially Missing Data: Introducing Multiple Imputation in Astrometric Analysis

Abstract

Partial measurements of relative position are a relatively common event during the observation of visual binary stars. However, these observations are typically discarded when estimating the orbit of a visual pair. In this article we present a novel framework to characterize the orbits from a Bayesian standpoint, including partial observations of relative position as an input for the estimation of orbital parameters. Our aim is to formally incorporate the information contained in those partial measurements in a systematic way into the final inference. In the statistical literature, an imputation is defined as the replacement of a missing quantity with a plausible value. To compute posterior distributions of orbital parameters with partial observations, we propose a technique based on Markov chain Monte Carlo with multiple imputation. We present the methodology and test the algorithm with both synthetic and real observations, studying the effect of incorporating partial measurements in the parameter estimation. Our results suggest that the inclusion of partial measurements into the characterization of visual binaries may lead to a reduction in the uncertainty associated to each orbital element, in terms of a decrease in dispersion measures (such as the interquartile range) of the posterior distribution of relevant orbital parameters. The extent to which the uncertainty decreases after the incorporation of new data (either complete or partial) depends on how informative those newly-incorporated measurements are. Quantifying the information contained in each measurement remains an open issue. 2 R. Claveria et al.observed, angular separations beyond the angular resolution of the telescope, etc. The latter case is arguably the most common source of partial information: in the vicinity of the periastron, the angular separation between the primary and secondary star (denoted by ρ hereafter) reaches its minimum values; if those values fall below the resolution threshold of the imaging device, then the angular separation is confined to a range (i.e., ρ ∈ (0, ρ max )) instead of being reduced to a single value (as occurs with regular, successfully resolved measurements), and usually no information about the position angle of the secondary with respect to the primary (denoted by ϑ hereafter) can be inferred either. In certain cases, databases of visual binary observations also report partial data of the form ρ = (0, ∞), ϑ = ϑ * , that is, the angular separation is missing but the position angle is well-defined. Many example of partially missing data (in ρ or ϑ) can be found by browsing either through the historical measurements database of the Washington Double Star Catalogue (WDS hereafter, Mason et al. (2001)) or the Catalog of Interferometric Measurements of Binary Stars (INT hereafter, Hartkopf et al. (2001)), both diligently curated by the Naval Observatory of the United States 1 .In a broader context, from population censuses to clinical research surveys, the presence of incomplete or missing data may occur in a wide ...

Orbits and masses of binaries from Speckle Interferometry at SOAR

“…The orbits have been computed using our MCMC code, duly described in Mendez et al (2017) and Claveria et al (2019). The quality of our final orbital elements is variable, ranging from tentative to good quality, depending on orbital coverage and overall data quality.…”

“…For our orbital calculations we have used a Bayesian MCMC orbital code with dimensionality reduction, whose implementation is described in detail in Mendez et al (2017) and Claveria et al (2019). The main motivation behind this approach is to exploit features that are inherent to these methods, namely (i) to provide realistic confidence limits to the derived orbital elements, and (ii) to generate posterior probability density functions (PDF hereafter) for each orbital element, as well as for the derived masses.…”

Data Science for Engineers: A Teaching Ecosystem

“…Lastly, for those of our graduates who pursue an academic career, the MSc thesis provides a fertile environment for theoretical research, where students join PhD students and postdocs to work under the close supervision of their mentors and, in most cases, successfully publish their findings (see, e.g., [29], [30], [31], [32], [33]).…”

Orbital elements and individual component masses from joint spectroscopic and astrometric data of double-line spectroscopic binaries

“…A good starting point for systematic surveys to determine stellar masses are all-sky catalogues which include identification of confirmed or suspected visual binaries, such as the Hipparcos catalogue (Lindegren et al 1997), as well as the more recent Gaia discoveries (Kervella et al 2019;El-Badry et al 2021;Brandt 2021); or spectroscopic binaries, such as the Geneva-Copenhagen spectroscopic survey (Nordström et al 2004). To this end, in 2014 we initiated a systematic campaign to complete or improve the observation of southern binaries from mainly the above catalogues (Mendez et al 2018), using the high-speed speckle camera HRCAM at the SOAR 4.1m telescope (Tokovinin & Cantarutti 2008;Tokovinin 2018); several publications have resulted from this effort, Gomez et al (2016); Mendez et al (2017); Claveria et al (2019); Docobo et al (2019); Mendez et al (2021); Villegas et al (2021); Gómez et al (2021). Considering that metal-poor binary systems are typically farther away and therefore fainter and/or more compact spatially, making them difficult objects for optical interferometry with 4m or smaller telescopes, in 2019 we also started a 1 Usually found only on young stars.…”