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Register a new userA robust statistical estimation of the basic parameters of single stellar populations. I. Method
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The colour-magnitude diagrams of resolved single stellar populations, such as open and globular clusters, have provided the best natural laboratories to test stellar evolution theory. Whilst a variety of techniques have been used to infer the basic properties of these simple populations, systematic uncertainties arise from the purely geometrical degeneracy produced by the similar shape of isochrones of different ages and metallicities. Here we present an objective and robust statistical technique which lifts this degeneracy to a great extent through the use of a key observable: the number of stars along the isochrone. Through extensive Monte Carlo simulations we show that, for instance, we can infer the four main parameters (age, metallicity, distance and reddening) in an objective way, along with robust confidence intervals and their full covariance matrix. We show that systematic uncertainties due to field contamination, unresolved binaries, initial or present-day stellar mass function are either negligible or well under control. This technique provides, for the first time, a proper way to infer with unprecedented accuracy the fundamental properties of simple stellar populations, in an easy-to-implement algorithm.
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RZ Cas is a short-period Algol-type system showing episodes of mass transfer and Delta Sct-like oscillations of its mass-gaining primary component. We analyse high-resolution spectra of RZ Cas that we obtained during a spectroscopic long-term monitoring lasting from 2001 to 2017. Spectrum analysis resulted in precise atmospheric parameters of both components, in particular in surface abundances below solar values. We find that the variation of orbital period is semi-regular and derive different characteristic timescales for different epochs of observation. We show that the radial velocity variations with orbital phase can be modelled when including two cool spots on the surface of the secondary component. The modelling leads to precise masses and separation of the components. The seasonal variation of several parameters, such as vsin(i), rotation-orbit synchronisation factor, strength of the spots on the cool companion, and orbital period, can be characterised by a common timescale of the order of nine years. We interpret the timescale of nine years as the magnetic activity cycle of the cool companion. In particular the behaviour of the dark spots on the cool companion leads us to the interpretation that this timescale is based on an 18-year magnetic dynamo cycle. We conclude that the mass-transfer rate is controlled by the variable depth of the Wilson depression in the magnetic spot around the Lagrangian point L1. In the result, based on available data, we observe a damped activity cycle of the star, starting with a high mass-transfer episode around 2001, followed by quiet periods in 2006 and 2009, slightly higher activity around 2013 and 2014, and again followed by quiet periods in 2015 and 2016. However, owing to missing data for years 2010 and 2011, we cannot exclude that a second high mass-transfer episode occurred within this time span.
Consider a setting with $N$ independent individuals, each with an unknown parameter, $p_i in [0, 1]$ drawn from some unknown distribution $P^star$. After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i sim text{Binomial}(t, p_i)$ per individual, our objective is to accurately estimate $P^star$. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large while the number of observations per individual is often limited. Our main result shows that, in the regime where $t ll N$, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large $N$, the MLE achieves the information theoretic optimal error bound of $mathcal{O}(frac{1}{t})$ for $t < clog{N}$, with regards to the earth movers distance (between the estimated and true distributions). More generally, in an exponentially large interval of $t$ beyond $c log{N}$, the MLE achieves the minimax error bound of $mathcal{O}(frac{1}{sqrt{tlog N}})$. In contrast, regardless of how large $N$ is, the naive plug-in estimator for this problem only achieves the sub-optimal error of $Theta(frac{1}{sqrt{t}})$.
We present a test of different error estimators for 2-point clustering statistics, appropriate for present and future large galaxy redshift surveys. Using an ensemble of very large dark matter LambdaCDM N-body simulations, we compare internal error estimators (jackknife and bootstrap) to external ones (Monte-Carlo realizations). For 3-dimensional clustering statistics, we find that none of the internal error methods investigated are able to reproduce neither accurately nor robustly the errors of external estimators on 1 to 25 Mpc/h scales. The standard bootstrap overestimates the variance of xi(s) by ~40% on all scales probed, but recovers, in a robust fashion, the principal eigenvectors of the underlying covariance matrix. The jackknife returns the correct variance on large scales, but significantly overestimates it on smaller scales. This scale dependence in the jackknife affects the recovered eigenvectors, which tend to disagree on small scales with the external estimates. Our results have important implications for the use of galaxy clustering in placing constraints on cosmological parameters. For example, in a 2-parameter fit to the projected correlation function, we find that the standard bootstrap systematically overestimates the 95% confidence interval, while the jackknife method remains biased, but to a lesser extent. The scatter we find between realizations, for Gaussian statistics, implies that a 2-sigma confidence interval, as inferred from an internal estimator, could correspond in practice to anything from 1-sigma to 3-sigma. Finally, by an oversampling of sub-volumes, it is possible to obtain bootstrap variances and confidence intervals that agree with external error estimates, but it is not clear if this prescription will work for a general case.
We develop a Bayesian model for globular clusters composed of multiple stellar populations, extending earlier statistical models for open clusters composed of simple (single) stellar populations (vanDyk et al. 2009, Stein et al. 2013). Specifically, we model globular clusters with two populations that differ in helium abundance. Our model assumes a hierarchical structuring of the parameters in which physical properties---age, metallicity, helium abundance, distance, absorption, and initial mass---are common to (i) the cluster as a whole or to (ii) individual populations within a cluster, or are unique to (iii) individual stars. An adaptive Markov chain Monte Carlo (MCMC) algorithm is devised for model fitting that greatly improves convergence relative to its precursor non-adaptive MCMC algorithm. Our model and computational tools are incorporated into an open-source software suite known as BASE-9. We use numerical studies to demonstrate that our method can recover parameters of two-population clusters, and also show model misspecification can potentially be identified. As a proof of concept, we analyze the two stellar populations of globular cluster NGC 5272 using our model and methods. (BASE-9 is available from GitHub: https://github.com/argiopetech/base/releases).
We analyze single-stellar-population (SSP) equivalent parameters for 50 local elliptical galaxies as a function of their structural parameters. These galaxies fill a two-dimensional plane in the four-dimensional space of [Z/H], log t, log $sigma$, and [E/Fe]. SSP age and velocity dispersion can be taken as the two independent parameters that specify a galaxys location in this ``hyperplane. The hyperplane can be decomposed into two sub-relations: (1) a ``Z-plane, in which [Z/H] is a linear function of log $sigma$ and log t; and (2) a relation between [E/Fe] and $sigma$ in which [E/Fe] is larger in high-$sigma$ galaxies. Cluster and field ellipticals follow the same hyperplane, but their ($sigma$,t) distributions within it differ. Nearly all cluster galaxies are old; the field ellipticals span a large range in SSP age. The tight Mg--$sigma$ relations of these ellipticals can be understood as two-dimensional projections of the metallicity hyperplane showing it edge-on; the tightness of these relations does not necessarily imply a narrow range of ages at fixed $sigma$. The relation between [E/Fe] and $sigma$ is consistent with a higher effective yield of Type II SNe elements at higher $sigma$. The Z-plane is harder to explain and may be a powerful clue to star formation in elliptical galaxies if it proves to be general. Present data favor a ``frosting model in which low apparent SSP ages are produced by adding a small frosting of younger stars to an older base population. If the frosting abundances are close to or slightly greater than the base population, simple two-component models run along lines of constant $sigma$ in the Z-plane, as required. This favors star formation from well-mixed pre-enriched gas rather than unmixed low-metallicity gas from an accreted object. (Abridged)
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