Stellar population synthesis with more degrees of freedom than observables

In order to derive the stellar population of a galaxy or a star cluster, it is a common practice to fit its spectrum by a combination of spectra extracted from a data base (e.g. a library of stellar spectra). If the data to be fitted are equivalent widths, the combination is a non-linear one and the problem of finding the 'best' combination of stars that fits the data becomes complex. It is probably because of this complexity that the mathematical aspects of the problem did not receive a satisfying treatment; the question of the uniqueness of the solution , for example, was left in uncertainty. In this paper we complete the solution of the problem by considering the underdetermined case where there are fewer equivalent widths to fit than stars in the data base (the overdetermined case was treated previously). The underdetermined case is interesting to consider because it leaves space for the addition of supplementary astrophysical constraints. In fact, it is shown in this paper that when a solution exists it is generally not unique. There are infinitely many solutions, all of them contained within a convex polyhedron in the solutions vector space. The vertices of this polyhedron are extremal solutions of the stellar population synthesis. If no exact solution exists, an approximate solution can be found using the method described for the overdetermined case. Also provided is an algorithm able to solve the problem numerically; in particular all the vertices of the polyhedron are found.     

The analysis of irregularly observed stochastic astronomical time-series – I. Basics of linear stochastic differential equations

The theory of low-order linear stochastic differential equations is reviewed. Solutions to these equations give the continuous time analogues of discrete time autoregressive time-series. Explicit forms for the power spectra and covariance functions of first- and second-order forms are given. A conceptually simple method is described for fitting continuous time autoregressive models to data. Formulae giving the standard errors of the parameter estimates are derived. Simulated data are used to verify the performance of the methods. Irregularly spaced observations of the two hydrogen-deficient stars FQ Aqr and NO Ser are analysed. In the case of FQ Aqr the best-fitting model is of second order, and describes a quasi-periodicity of about 20 d with an e-folding time of 3.7 d. The NO Ser data are best fitted by a first-order model with an e-folding time of 7.2 d.     

Estimating the spectral indices of correlated astrophysical foregrounds by a second-order statistical approach

We present the first tests of a new method, the correlated component analysis (CCA) based on second-order statistics, to estimate the mixing matrix, a key ingredient to separate astrophysical foregrounds superimposed to the Cosmic Microwave Background (CMB). In the present application, the mixing matrix is parametrized in terms of the spectral indices of Galactic synchrotron and thermal dust emissions, while the free–free spectral index is prescribed by basic physics, and is thus assumed to be known. We consider simulated observations of the microwave sky with angular resolution and white stationary noise at the nominal levels for the Planck satellite, and realistic foreground emissions, with a position-dependent synchrotron spectral index. We work with two sets of Planck frequency channels: the low-frequency set, from 30 to 143 GHz, complemented with the Haslam 408 MHz map, and the high-frequency set, from 217 to 545 GHz. The concentration of intense free–free emission on the Galactic plane introduces a steep dependence of the spectral index of the global Galactic emission with Galactic latitude, close to the Galactic equator. This feature makes difficult for the CCA to recover the synchrotron spectral index in this region, given the limited angular resolution of Planck , especially at low frequencies. A cut of a narrow strip around the Galactic equator  (| b | < 3°)  , however, allows us to overcome this problem. We show that, once this strip is removed, the CCA allows an effective foreground subtraction, with residual uncertainties inducing a minor contribution to errors on the recovered CMB power spectrum.     

Multimass schemes for collisionless N-body simulations

We present a general scheme for constructing Monte Carlo realizations of equilibrium, collisionless galaxy models with known distribution function (DF) f ₀. Our method uses importance sampling to find the sampling DF f _s that minimizes the mean-square formal errors in a given set of projections of the DF f ₀. The result is a multimass N -body realization of the galaxy model in which 'interesting' regions of phase space are densely populated by lots of low-mass particles, increasing the effective N there, and less interesting regions by fewer, higher mass particles.
As a simple application, we consider the case of minimizing the shot noise in estimates of the acceleration field for an N -body model of a spherical Hernquist model. Models constructed using our scheme easily yield a factor of ∼100 reduction in the variance at the central acceleration field when compared to a traditional equal-mass model with the same number of particles. When evolving both models with a real N -body code, the diffusion coefficients in our model are reduced by a similar factor. Therefore, for certain types of problems, our scheme is a practical method for reducing the two-body relaxation effects, thereby bringing the N -body simulations closer to the collisionless ideal.     

Massive lossless data compression and multiple parameter estimation from galaxy spectra

We present a method for radical linear compression of data sets where the data are dependent on some number M of parameters. We show that, if the noise in the data is independent of the parameters, we can form M linear combinations of the data which contain as much information about all the parameters as the entire data set, in the sense that the Fisher information matrices are identical; i.e. the method is lossless. We explore how these compressed numbers fare when the noise is dependent on the parameters, and show that the method, though not precisely lossless, increases errors by a very modest factor. The method is general, but we illustrate it with a problem for which it is well-suited: galaxy spectra, the data for which typically consist of ∼10³ fluxes, and the properties of which are set by a handful of parameters such as age, and a parametrized star formation history. The spectra are reduced to a small number of data, which are connected to the physical processes entering the problem. This data compression offers the possibility of a large increase in the speed of determining physical parameters. This is an important consideration as data sets of galaxy spectra reach 10⁶ in size, and the complexity of model spectra increases. In addition to this practical advantage, the compressed data may offer a classification scheme for galaxy spectra which is based rather directly on physical processes.     

Time-dependent three-dimensional spectrum synthesis for Type Ia supernovae

A Monte Carlo code ( artis ) for modelling time-dependent three-dimensional spectral synthesis in chemically inhomogeneous models of Type Ia supernova ejecta is presented. Following the propagation of γ-ray photons, emitted by the radioactive decay of the nucleosynthesis products, energy is deposited in the supernova ejecta and the radiative transfer problem is solved self-consistently, enforcing the constraint of energy conservation in the comoving frame. Assuming a photoionization-dominated plasma, the equations of ionization equilibrium are solved together with the thermal balance equation adopting an approximate treatment of excitation. Since we implement a fully general treatment of line formation, there are no free parameters to adjust. Thus, a direct comparison between synthetic spectra and light curves, calculated from hydrodynamic explosion models, and observations is feasible. The code is applied to the well-known W7 explosion model and the results tested against other studies. Finally, the effect of asymmetric ejecta on broad-band light curves and spectra is illustrated using an elliptical toy model.     

Data boundary fitting using a generalized least-squares method

In many astronomical problems one often needs to determine the upper and/or lower boundary of a given data set. An automatic and objective approach consists in fitting the data using a generalized least-squares method, where the function to be minimized is defined to handle asymmetrically the data at both sides of the boundary. In order to minimize the cost function, a numerical approach, based on the popular downhill simplex method, is employed. The procedure is valid for any numerically computable function. Simple polynomials provide good boundaries in common situations. For data exhibiting a complex behaviour, the use of adaptive splines gives excellent results. Since the described method is sensitive to extreme data points, the simultaneous introduction of error weighting and the flexibility of allowing some points to fall outside of the fitted frontier, supplies the parameters that help to tune the boundary fitting depending on the nature of the considered problem. Two simple examples are presented, namely the estimation of spectra pseudo-continuum and the segregation of scattered data into ranges. The normalization of the data ranges prior to the fitting computation typically reduces both the numerical errors and the number of iterations required during the iterative minimization procedure.     

Bayesian photometric redshifts with empirical training sets

We combine in a single framework the two complementary benefits of  χ²  template fits and empirical training sets used e.g. in neural nets:  χ²  is more reliable when its probability density functions (PDFs) are inspected for multiple peaks, while empirical training is more accurate when calibration and priors of query data and training set match. We present a  χ²  empirical method that derives PDFs from empirical models as a subclass of kernel regression methods, and apply it to the Sloan Digital Sky Survey Data Release 5 sample of >75 000 quasi-stellar objects, which is full of ambiguities. Objects with single-peak PDFs show <1 per cent outliers, rms redshift errors <0.05 and vanishing redshift bias. At   z > 2.5  , these figures are two times better. Outliers result purely from the discrete nature and limited size of the model, and rms errors are dominated by the intrinsic variety of object colours. PDFs classed as ambiguous provide accurate probabilities for alternative solutions and thus weights for using both solutions and avoiding needless outliers. E.g. the PDFs predict 78.0 per cent of the stronger peaks to be correct, which is true for 77.9 per cent of them. Redshift incompleteness is common in faint spectroscopic surveys and turns into a massive undetectable outlier risk above other performance limitations, but we can quantify residual outlier risks stemming from size and completeness of the model. We propose a matched  χ²  error scale for noisy data and show that it produces correct error estimates and redshift distributions accurate within Poisson errors. Our method can easily be applied to future large galaxy surveys, which will benefit from the reliability in ambiguity detection and residual risk quantification.     

Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Determination of orbital parameters from observations is formally a nonlinear inverse problem for solving which evidently nonlinear methods are required. Meanwhile, an accompanying stage in solving the inverse problem is the evaluation of parametric accuracy to which, however, linear methods are conventionally applied. This is quite justified if parametric errors caused by observation errors are rather small, otherwise this is not at all since the nonlinearity of the inverse problem can be considerable to influence on the evaluations of parametric accuracy especially when the observations are very few. With the advent of quick-operating and multiprocessor computers, recently one tends to employ statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations. In the paper are just discussed the methods designed specially for nonlinear statistic simulation of virtual parameter values. Their efficiency is investigated in application to estimating uncertainties in the orbit of Jovian satellite S/2003 J04 whose orbital parameters are ill-determined owing to scanty available observations. Indices of nonlinearity are introduced for making decision in the choice between linear and nonlinear methods.     

Cosmological systematics beyond nuisance parameters: form-filling functions

In the absence of any compelling physical model, cosmological systematics are often misrepresented as statistical effects and the approach of marginalizing over extra nuisance systematic parameters is used to gauge the effect of the systematic. In this article, we argue that such an approach is risky at best since the key choice of function can have a large effect on the resultant cosmological errors.
As an alternative we present a functional form-filling technique in which an unknown, residual, systematic is treated as such. Since the underlying function is unknown, we evaluate the effect of every functional form allowed by the information available (either a hard boundary or some data). Using a simple toy model, we introduce the formalism of functional form filling. We show that parameter errors can be dramatically affected by the choice of function in the case of marginalizing over a systematic, but that in contrast the functional form-filling approach is independent of the choice of basis set.
We then apply the technique to cosmic shear shape measurement systematics and show that a shear calibration bias of  | m ( z )| ≲ 10⁻³ (1 + z )^0.7  is required for a future all-sky photometric survey to yield unbiased cosmological parameter constraints to per cent accuracy.
A module associated with the work in this paper is available through the open source icosmo code available at http://www.icosmo.org .     

Inference of relativistic electron spectra from measurements of inverse compton radiation

The inference of relativistic electron spectra from spectral measurement of inverse Compton radiation is discussed for the case where the background photon spectrum is a Planck function. The problem is formulated in terms of an integral transform that relates the measured spectrum to the unknown electron distribution. A general inversion formula is used to provide a quantitative assessment of the information content of the spectral data. It is shown that the observations must generally be augmented by additional information if anything other than a rudimentary two or three parameter model of the source function is to be derived. It is also pointed out that since a similar equation governs the continuum spectra emitted by a distribution of black-body radiators, the analysis is relevant to the problem of stellar population synthesis from galactic spectra.     

A new method for determining the sensitivity of X-ray imaging observations and the X-ray number counts

We present a new method for determining the sensitivity of X-ray imaging observations, which correctly accounts for the observational biases that affect the probability of detecting a source of a given X-ray flux, without the need to perform a large number of time-consuming simulations. We use this new technique to estimate the X-ray source counts in different spectral bands (0.5–2, 0.5–10, 2–10 and 5–10 keV) by combining deep pencil-beam and shallow wide-area Chandra observations. The sample has a total of 6295 unique sources over an area of  11.8 deg²  and is the largest used to date to determine the X-ray number counts. We determine, for the first time, the break flux in the 5–10 keV band, in the case of a double power-law source count distribution. We also find an upturn in the 0.5–2 keV counts at fluxes below about  6 × 10⁻¹⁷ erg s⁻¹ cm⁻²  . We show that this can be explained by the emergence of normal star-forming galaxies which dominate the X-ray population at faint fluxes. The fraction of the diffuse X-ray background resolved into point sources at different spectral bands is also estimated. It is argued that a single population of Compton thick active galactic nuclei (AGN) cannot be responsible for the entire unresolved X-ray background in the energy range 2–10 keV.     

Peranso – Light curve and period analysis software

A time series is a sample of observations of well‐defined data points obtained through repeated measurements over a certain time range. The analysis of such data samples has become increasingly important not only in natural science but also in many other fields of research. Peranso offers a complete set of powerful light curve and period analysis functions to work with large astronomical data sets. Substantial attention has been given to ease‐of‐use and data accuracy, making it one of the most productive time series analysis software available. In this paper, we give an introduction to Peranso and its functionality. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Foregrounds for redshifted 21-cm studies of reionization: Giant Meter Wave Radio Telescope 153-MHz observations

Foreground subtraction is the biggest challenge for future redshifted 21-cm observations to probe reionization. We use a short Giant Meter Wave Radio Telescope (GMRT) observation at 153 MHz to characterize the statistical properties of the background radiation across ∼1° to subarcmin angular scales, and across a frequency band of 5 MHz with 62.5 kHz resolution. The statistic we use is the visibility correlation function, or equivalently the angular power spectrum   C _l   . We present the results obtained from using relatively unsophisticated, conventional data calibration procedures. We find that even fairly simple-minded calibration allows one to estimate the visibility correlation function at a given frequency   V ₂( U , 0)  . From our observations, we find that   V ₂( U , 0)  is consistent with foreground model predictions at all angular scales except the largest ones probed by our observations where the model predictions are somewhat in excess. On the other hand, the visibility correlation between different frequencies  κ( U , Δν)  seems to be much more sensitive to calibration errors. We find a rapid decline in  κ( U , Δν)  , in contrast with the prediction of less than 1 per cent variation across 2.5 MHz. In this case, however, it seems likely that a substantial part of the discrepancy may be due to limitations of data reduction procedures.     

A Roche model for uniformly rotating rings

A Roche model for describing uniformly rotating rings is presented, and the results are compared with the numerical solutions to the full problem for polytropic rings. In the thin ring limit, the surfaces of constant pressure including the surface of the ring itself are given in analytical terms, even in the mass-shedding case.     

The analysis of indexed astronomical time series – VI. Covariances of amplitude ratios and phase differences estimated from multicolour photometry of a pulsating star

One of the tools used to identify the pulsation modes of stars is a comparison of the amplitudes and phases as observed photometrically at different wavelengths. Proper application of the method requires that the errors on the measured quantities, and the correlations between them, be known (or at least estimated). It is assumed that contemporaneous measurements of the light intensity of a pulsating star are obtained in several wavebands. It is also assumed that the measurements are regularly spaced in time, although there may be missing observations. The amplitude and phase of the pulsation are estimated separately for each of the wavebands, and amplitude ratios and phase differences are calculated. A general scheme for estimating the covariance matrix of the amplitude ratios and phase differences is described. The first step is to fit a time series to the residuals after pre-whitening the observations by the best-fitting sinusoid. The residuals are then cross-correlated to study the interdependence between the errors in the different wavebands. Once the multivariate time-series structure can be modelled, the covariance matrix can be found by bootstrapping. An illustrative application is described in detail.     

The analysis of indexed astronomical time series V. Fitting sinusoids to high-speed photometry

The estimation of the frequency, amplitude and phase of a sinusoid from observations contaminated by correlated noise is considered. It is assumed that the observations are regularly spaced, but may suffer missing values or long time stretches with no data. The typical astronomical source of such data is high-speed photoelectric photometry of pulsating stars. The study of the observational noise properties of nearly 200 real data sets is reported: noise can almost always be characterized as a random walk with superposed white noise. A scheme for obtaining weighted non-linear least-squares estimates of the parameters of interest, as well as standard errors of these estimates, is described. Simulation results are presented for both complete and incomplete data. It is shown that, in finite data sets, results are sensitive to the initial phase of the sinusoid.     

Construction of confidence regions in the problem of asteroid orbit determination

The factors required for estimation of the accuracy of the confidence region construction in the problem of asteroid orbit determination are considered. Blunders and large systematic errors occurring in asteroid observations increase the sizes of confidence regions and cause their noticeable shift in the space of determined parameters. We present the factors that, in addition to analysis of discrepancies (O–C), provide an opportunity to estimate the efficiency of screening observations containing gross systematic errors. The developed factors have been tested for efficiency using simulated observations. The observations have been simulated by parameters set by us and assumed true. It is shown how the sizes of systematic errors and the number of observations with these errors influence the results of screening. All calculations have been performed within the Keplerian model of asteroid motion.     

Bias properties of logarithmic and asinh magnitudes at very low flux levels

The observation of flux sources near the limit of detection requires a careful evaluation of possible biases in magnitude determination. Both the traditional logarithmic magnitudes and the recently proposed inverse hyperbolic sine (asinh) magnitudes are considered. Formulae are derived for three different biasing mechanisms: the statistical spread of the observed flux values arising from e.g. measurement error; the dependence of these errors on the true flux; and the dependence of the observing probability on the true flux. As an example of the results, it is noted that biases at large signal-to-noise ratios R , at which the two types of magnitude are similar, are of the order of −( p +1)/ R ², where the exponent p parametrizes a power-law dependence of the probability of observation on the true flux.     

Empirical calibration of the near-infrared Ca ii triplet – I. The stellar library and index definition

A new stellar library at the near-IR spectral region developed for the empirical calibration of the Ca  ii triplet and stellar population synthesis modelling is presented. The library covers the range λλ 8348–9020 at 1.5-Å (FWHM) spectral resolution, and consists of 706 stars spanning a wide range in atmospheric parameters. We have defined a new set of near-IR indices, CaT*, CaT and PaT, which mostly overcome the limitations of previous definitions, the former being specially suited for the measurement of the Ca  ii triplet strength corrected for the contamination from Paschen lines. We also present a comparative study of the new and the previous Ca indices, as well as the corresponding transformations between the different systems. A thorough analysis of the sources of index errors and the procedure to calculate them is given. Finally, index and error measurements for the whole stellar library are provided together with the final spectra.