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.     

Fitting power-law distributions to data with measurement errors

If X , which follows a power-law distribution, is observed subject to Gaussian measurement error e , then   X + e   is distributed as the convolution of the power-law and Gaussian distributions. Maximum-likelihood estimation of the parameters of the two distributions is considered. Large-sample formulae are given for the covariance matrix of the estimated parameters, and implementation of a small-sample method (the jackknife) is also described. Other topics dealt with are tests for goodness of fit of the posited distribution, and tests whether special cases (no measurement errors or an infinite upper limit to the power-law distribution) may be preferred. The application of the methodology is illustrated by fitting convolved distributions to masses of giant molecular clouds in M33 and the Large Magellanic Cloud (LMC), and to H  i cloud masses in the LMC.     

On the frequency and amplitude variations of the δ Scuti star CD−24 7599 (=XX Pyx)

We present 132 h of new time-series photometric observations of the δ Scuti star CD−24 7599 acquired during 86 nights from 1993 to 1996 to study its frequency and amplitude variations. By using all published observations we demonstrate that the three dominating pulsation modes of the star can change their photometric amplitudes within one month at certain times, while the amplitudes can remain constant within the measurement errors at other times. CD−24 7599 also exhibits frequency variations, which do not show any correspondence between the different modes.   The typical time-scale for the amplitude variations is found to be several hundred days, which is of the same order of magnitude as the inverse linear growth rates of a selected model. We find no evidence for periodic amplitude modulation of two of the investigated modes ( f ₂ and f ₃), but f ₁ may exhibit periodic modulation. The latter result could be spurious and requires confirmation. The observed frequency variations may either be continuous or reflect sudden frequency jumps. No evidence for cyclical period changes is obtained.   We exclude precession of the pulsation axis and oblique pulsation for the amplitude variations. Beating of closely spaced frequencies cannot explain the amplitude modulations of two of the modes, while it is possible for the third. Evolutionary effects, binarity, magnetic field changes or avoided crossings cannot be made responsible for the observed period changes. Only resonance between different modes may be able to explain the observations. However, at this stage a quantitative comparison is not possible. More observations, especially data leading to a definite mode identification and further measurements of the temporal behaviour of the amplitudes and frequencies of CD−24 7599, are required.     

UBVRIJH photometry of two new luminous δ Scuti stars and the discovery of δ Scuti pulsation in the most evolved Ap star known

Time-series photometry of the Hipparcos variable stars HD 199434 and 21190 is reported. Both stars are pulsators of the δ Scuti type. Reclassifications of the MK types of the stars, based on new spectrograms, are given. HD 21190 is found to be F2III SrEuSi:, making it the most evolved Ap star known. Its Strömgren photometric indices support the peculiar spectral type. It is also one of the most evolved δ Scuti stars known. Its combined Ap– δ Scuti nature makes it an important test of models of pulsation in peculiar stars recently developed by Turcotte et al., although it is more extreme than any model they examined. Physical parameters of both stars are estimated from Strömgren and H β photometry, and Hipparcos absolute magnitudes. We attempt mode identifications based on amplitude ratios and phase differences from our photometry. The dominant pulsation of HD 21190 may be an overtone radial mode. The model fits for HD 199434 are even less satisfactory, but favour an ℓ=2 mode. Given the good quality and wavelength coverage of our data, the poor results from the application of the photometric theory of mode identification may call into question the use of that technique.     

Spectroscopy of the roAp star α Cir — I. Velocities of Hα and metal lines

We present radial velocity measurements of the rapidly oscillating Ap (roAp) star α Cir, obtained from dual-site observations with medium-dispersion spectrographs. The amplitude and phase of the principal pulsation mode vary significantly, depending on which line is being measured. The amplitude is observed to be as high as 1000 m s⁻¹ in some wavelength bands, despite a previous upper limit of 36 m s⁻¹. Furthermore, some lines are apparently pulsating in anti-phase with others. We suggest this indicates a high-overtone standing wave with a velocity node in the atmosphere of the star.     

Numerical solutions of the equation concerning small adiabatic oscillations of pulsating components in double star systems

The equation governing small adiabatic radial oscillations for pulsating components in close binary stars modelled by tidally and rotationally distorted Roche geometry is solved numerically. With assumed initial conditions, solutions for systems with different mass ratios are presented. The changes in relative wave amplitude with various parameters are shown. The variation of the ratio of the pulsation frequencies of distorted to undistorted stars for given mass ratio of the binary systems is also investigated. Observational evidence is examined by using two data sets that show the modelled effects, by taking into account likely practical factors. The results show that the measured frequencies and amplitudes of surface waves can vary slightly for distorted stars in comparison to undistorted ones (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Time-dependent convection seismic study of five γ Doradus stars

We apply for the first time the time-dependent convection (TDC) treatment of Gabriel and Grigahcène et al. to the photometric mode identification in γ Doradus (γ Dor) stars. We consider the influence of this treatment on the theoretical amplitude ratios and phase differences. Comparison with the observed amplitudes and phases of the stars γ Dor, 9 Aurigae, HD 207223 = HR 8330, HD 12901 and 48501 is presented and enables us to identify the degree ℓ of the pulsation modes for four of them. We also determine the mode stability for different models of these stars. We show that our TDC models agree better with observations than with frozen convection models. Finally, we compare the results obtained with different values of the mixing-length parameter α.     

KPD 0629–0016, a slowly pulsating hot subdwarf star

The results of nine CCD photometric observing runs on KPD 0629–0016 are presented. During six of the runs measurements were obtained alternately through B and V filters. Four periodicities, for which there is good agreement between the frequencies identified in the B and V data, were extracted: these lie in the range 46–81 min. A fifth lower frequency appears to be definitely present, but is very uncertain due to aliasing. The ratios of the mode amplitudes measured in B and V , and the phase differences between variations in the two colours, are compatible with pulsation theory.     

Patterns of aperiodic pulsation

Techniques for deriving amplitude equations for stellar pulsation are outlined. For the simplest such equations with multiple instabilities, the derivation of a map for the patterns of pulsation phases is described. This map gives the time between two successive maxima of pulsation in terms of the time between the previous pair, under suitable conditions. The phase differences can be regular, chaotic or hyperchaotic.     

The period and amplitude changes of Polaris (α UMi) from 2003 to 2007 measured with SMEI

We present an analysis of 4.5 yr of high precision (0.1 per cent) space-based photometric measurements of the Cepheid variable Polaris, obtained by the broad-band Solar Mass Ejection Imager (SMEI) instrument on board the Coriolis satellite. The data span from 2003 April to 2007 October, with a cadence of 101 min and a fill factor of 70 per cent. We have measured the mean peak-to-peak amplitude across the whole set of observations to be 25 mmag. There is, however, a clear trend that the size of the oscillations has been increasing during the observations, with peak-to-peak variations less than 22 mmag in early 2003, increasing to around 28 mmag by 2007 October, suggesting that the peak-to-peak amplitude is increasing at a rate of  1.39 ± 0.12 mmag yr⁻¹  . Additionally, we have combined our new measurements with archival measurements to measure a rate of period change of  4.90 ± 0.26 s yr⁻¹  over the last 50 yr. However, there is some suggestion that the period of Polaris has undergone a recent decline, and combined with the increased amplitude, this could imply evolution away from an overtone pulsation mode into the fundamental or a double pulsation mode depending on the precise mass of Polaris.     

Maximum likelihood estimates of the two- and three-dimensional power spectra of the APM Galaxy Survey

We estimate the two- and three-dimensional power spectra, P ₂( K ) and P ₃( k ), of the galaxy distribution by applying a maximum likelihood estimator to pixel maps of the APM Galaxy Survey. The analysis provides optimal estimates of the power spectra and of their covariance matrices if the fluctuations are assumed to be Gaussian. Our estimates of P ₂( K ) and P ₃( k ) are in good agreement with previous work, but we find that the errors at low wavenumbers have been underestimated in some earlier studies. If the galaxy power spectrum is assumed to have the same shape as the mass power spectrum, then the APM maximum likelihood P ₃( k ) estimates at k ≤0.19  h  Mpc⁻¹ constrain the amplitude and shape parameter of a scale-invariant CDM model to lie within the 2 σ ranges 0.74≤( σ ₈)_g≤1.28 and 0.06≤Γ≤0.46 . Using the Galactic extinction estimates of Schlegel, Finkbeiner & Davis, we show that Galactic obscuration has a negligible effect on galaxy clustering over most of the area of the APM Galaxy Survey.     

Unbiased reconstruction of the large-scale structure

We present a new unbiased minimal variance (UMV) estimator for the purpose of reconstructing the large-scale structure of the Universe from noisy, sparse and incomplete data. Similar to the Wiener filter (WF), the UMV estimator is derived by requiring the linear minimal variance solution given the data and an assumed a priori model specifying the underlying field covariance matrix. However, unlike the WF, the minimization is carried out with the added constraint of an unbiased reconstructed mean field. The new estimator does not necessitate a noise model to estimate the underlying field; however, such a model is required for evaluating the errors at each point in space. The general application of the UMV estimator is to predict the values of the reconstructed field in unsampled regions of space (e.g. interpolation in the unobserved Zone of Avoidance), and to dynamically transform from one measured field to another (e.g. inversion of radial peculiar velocities to over-densities). Here, we provide two very simple applications of the method. The first is to recover a 1D signal from noisy, convolved data with gaps, for example CMB time-ordered data. The second application is a reconstruction of the density and 3D peculiar velocity fields from mock SEcat galaxy peculiar velocity catalogues.     

Phasing segmented mirrors using defocused images at visible wavelengths

Plans for future optical telescopes of diameter more than 10 m are based on segmented mirrors, made up of hundreds or even thousands of segments. A challenge for these telescopes is the alignment in piston (cophasing) where phase differences between individual segments have to be reduced to a small fraction of the observing wavelength in order to avoid degradation of image quality.
Based on the phase discontinuity sensing method used at the Keck telescopes to measure small piston errors using infrared wavelengths, we develop a new method that allows fast high-precision measurements of large piston errors even at visible wavelengths.     

The rapidly oscillating Ap star HD 99563 and its distorted dipole pulsation mode

We undertook a time-series photometric multisite campaign for the rapidly oscillating Ap (roAp) star HD 99563 and also acquired mean light observations over four seasons. The pulsations of the star, which show flatter light maxima than minima, can be described with a frequency quintuplet centred on 1557.653 μHz and some first harmonics of it. The amplitude of the pulsation is modulated with the rotation period of the star that we determine with 2.91179 ± 0.00007 d from the analysis of the stellar pulsation spectrum and of the mean light data. We break up the distorted oscillation mode into its pure spherical harmonic components and find it is dominated by the ℓ= 1 pulsation, and also has a notable ℓ= 3 contribution, with weak ℓ= 0 and 2 components. The geometrical configuration of the star allows us to see both pulsation poles for about the same amount of time; HD 99563 is only the fourth roAp star for which both pulsation poles are seen and only the third where the distortion of the pulsation modes has been modelled. We point out that HD 99563 is very similar to the well-studied roAp star HR 3831. Finally, we note that the visual companion of HD 99563 is located in the δ Scuti instability strip and may thus show pulsation. We show that if the companion was physical, the roAp star would be a 2.03-M_⊙, object, seen at a rotational inclination of 44°, which then predicts a magnetic obliquity     .     

Cyclic variations in the angular diameter of χ Cygni

We report the direct detection of cyclic diameter variations in the Mira variable χ Cygni. Interferometric observations made between 1997 July and 1999 September, using the Cambridge Optical Aperture Synthesis Telescope (COAST) and the William Herschel Telescope (WHT), indicate periodic changes in the apparent angular diameter at a wavelength of 905 nm, with amplitude 45 per cent of the smallest value. The star appears largest at minimum light. Measurements made at a wavelength of 1.3 μm over the same period suggest much smaller size changes. This behaviour is consistent with a model in which most of the apparent diameter variation at 905 nm is caused by a large increase in the opacity of the outer atmospheric layers (which is mostly owing to titanium oxide) near minimum light, rather than by physical motions of the photosphere. The 1.3-μm waveband is relatively uncontaminated by TiO, and so much smaller size changes would be expected in this band. The latest non-linear pulsational models predict maximum physical size close to maximum light, and increases in opacity near minimum light that are too small to reproduce the diameter variation seen at 905 nm. This suggests either that the phase-dependence of the model pulsation is incorrect, or that the opacities in the models are underestimated. Future interferometric monitoring in uncontaminated near-infrared wavebands should resolve this question.     

Resolving the pulsations of the subdwarf B star KPD 2109 + 4401

We present the results of extensive time-series photometry of the pulsating subdwarf B star KPD 2109 + 4401. Our data set consists of 29 data runs with a total length of 182.6 h over 31 d, collected at five observatories in 2004. These data comprised high signal-to-noise ratio observations acquired with larger telescopes and wider time-coverage observations obtained with smaller telescopes. They are sufficient to resolve the pulsation structure to 0.4 μHz and are the most extensive data set for this star to date. With these data, we identify eight pulsation frequencies extending from 4701 to 5481 μHz, corresponding to periods of 182–213 s. The pulsation frequencies and their amplitudes are examined over several time-scales with some frequencies showing amplitude variability.     

Irradiation pressure effects in close binary systems

We test an analytic model for the two-point correlations of galaxy clusters in redshift space using the Hubble volume N -body simulations. The correlation function of clusters shows no enhancement along the line of sight, owing to the lack of any virialized structures in the cluster distribution. However, the distortion of the clustering pattern arising from coherent bulk motions is clearly visible. The distribution of cluster peculiar motions is well described by a Gaussian, except in the extreme high-velocity tails. The simulations produce a small but significant number of clusters with large peculiar motions. The form of the redshift-space power spectrum is strongly influenced by errors in measured cluster redshifts in extant surveys. When these errors are taken into account, the model reproduces the power spectrum recovered from the simulation to an accuracy of 15 per cent or better over a decade in wavenumber. We compare our analytic predictions with the power spectrum measured from the APM cluster redshift survey. The cluster power spectrum constrains the amplitude of density fluctuations, as measured by the linear rms variance in spheres of radius 8  h ⁻¹ Mpc, denoted by σ ₈. When combined with the constraints on σ ₈ and the density parameter Ω derived from the local abundance of clusters, we find a best-fitting cold dark matter model with     and     , for a power spectrum shape that matches that measured for galaxies. However, for the best-fitting value of Ω and given the value of Hubble's constant from recent measurements, the assumed shape of the power spectrum is incompatible with the most readily motivated predictions from the cold dark matter paradigm.     

The Planck Surveyor mission and gravitational lenses

An extremely sensitive all-sky survey will be carried out in the millimetre/submillimetre waveband by the forthcoming ESA mission Planck Surveyor . The main scientific goal of the mission is to make very accurate measurements of the spatial power spectrum of primordial anisotropies in the cosmic microwave background radiation; however, hundreds of thousands of distant dusty galaxies and quasars will also be detected. These sources are much more likely to be gravitationally lensed by intervening galaxies compared with sources discovered in surveys in other wavebands. Here the number of lenses expected in the survey is estimated, and techniques for discriminating between lensed and unlensed sources are discussed. A practical strategy for this discrimination is presented, based on exploiting the remarkable sensitivity and resolving power of large ground-based millimetre/submillimetre-wave interferometer arrays. More than a thousand gravitational lenses could be detected: a sample that would be an extremely valuable resource in observational cosmology.     

Mode Identification from Line Profiles using the Direct Fitting Technique

The method of moments and the direct fitting method are the onlyspectroscopic methods of mode identification which allow a determination ofall pulsational parameters. The pulsation parameters are required to predictthe light amplitude and phase which can be important discriminants in modeidentification. The direct fitting method has several advantages over themethod of moments. It is not restricted to low spherical harmonic degree or form of the eigenfunction and is not sensitive to the placement of the continuum. In the last few years the method has been applied to several different types of stars. We briefly describe the method and give someexamples of its application.