Period analyses without

Transcription

1 Period analyses without O-C diagrams Zdeněk Mikulášek, Miloslav Zejda, Jan Janík, Masaryk University, Brno, Czech Republic IAUS 282: From Interacting Binaries to Exoplanets: Essential Modeling Tools, Tatranská Lomnica, July 2011

2 Why to analyse variable stars periods? Most of variable stars show cyclic or periodic light variations. Light (spectroscopic and other) changes are repeating with one period P or with multiple periods. The basic mechanisms of the periodic VS variability: Rotation solid body, differential Orbital motion(s) other stars, planets Apsidal motion Pulsation(s) radial, non-radial Oscillations

3 Why to analyse variable stars periods? Most of variable stars show cyclic or periodic light variations. Light (spectroscopic and other) changes are repeating with one period P or with multiple periods. The period/s correspond/s to period of mechanisms of VS variability. Basic period analysis reliable ephemeris + modelling of 1-st order effects Fine period analyses additional information about the VS physics Further bodies in the system (stars, planets) Mass exchange between components of interacting binaries Spin-orbital interactions in the system Inner stellar structure + GR testing apsidal motion AM loss through stellar winds Gravitational waves Oscillations in rotation of some CP stars Differential rotation of cool stars These period variations are delicate a fine tool for their analysis is needed!

4 The role of O-C diagrams Practically all period analyses were done using O-CDs or their modifications. O the observed time of the LC extreme, C the time of the same extreme calculated by means of an ephemeris (as a rule linear). Historical O-times and their errors can be taken from the literature or specialized databases. Interpretation of O-CD: fit of data by weighted LSM. Aim: to determine parameters of the modelling with some astrophysical meaning and to estimate their uncertainties by standard wlsm techniques. The weights of individual measurements ~ δ -2 (O). Scatter used to be several times larger then the expected warning indication

5 Afflictions of standard O-C diagrams The credibility of obtained astrophysical information strictly depends namely on the reliability of O times and their uncertainties. 1. Moments of extrema of observed time series are always bad defined even small scatter can strongly influence the individual O determinations. Much better info represents the phase shift of an observed LC in respect to a template LC. 2. The techniques of standard extrema times determinations (Kwee-van Woerden etc.) are almost irreproducible and give unreliable results. The fitted profiles do not correspond to the true LC profiles e.g. minima in EBs are not parabolas! 3. If the fitted LC does not correspond to the true one it cannot be used standard LSM approach to the estimation of uncertainties of parameters. Consequently, the majority of given uncertainties are fully unrealistic do not use them! 4. The most serious imperfection: Standard methods do not respect the fact that the treated stars are periodic. It causes grave loss of information the real scatter of such values is standardly 2.5 times larger than the scatter of properly processed data.

6 Afflictions of standard O-C diagrams The credibility of obtained astrophysical information strictly depends namely on the reliability of O times and their uncertainties. Standard O-C diagrams are not very reliable tools for period analyses, they have their specific limitations. What shall we do? How to do the reliability of demanding period analyzes better? A) We can improve their input reliable O values using the periodicity of LC. B) Much better: We can cancel the O-C inter-step completely!

7 M 0, P

8 M 0, P

9 Afflictions of standard O-C diagrams The credibility of obtained astrophysical information strictly depends namely on the reliability of O times and their uncertainties. Standard O-C diagrams are not very reliable tools for period analyses, they have their specific limitations. What shall we do? How to do the reliability of demanding period analyses better? A) We can improve their input reliable O values using the periodicity of LC. B) Much better: We can cancel the O-C inter-step completely! We can analyse period variations directly using original observations of all kind! The need of original data is the only limitation of the method in several cases we are unable to trace back them and we have to content only with the A approach.

10 Direct analyses and modeling of period variations The basic observational data time series {t i, y i, w i }. Observed (light) variations y can be modeled by a template periodic phase curve F(ϑ, a). ϑ(t,b) is the time growing phase function (IP - epoch, FP - phase). The rigorous definition by means of the time variable period P(t,b): t dt dτ = P( t, b) ϑ( t, b) = d ϑ P ( τ, b) Models consists of two parts: F(ϑ,, a) and ϑ(t,, b). Both sets of parameters can be found by the standard weighted LSM or a robust regression diminishing the influence of frequent outliers. The adequacy of modeling is given by the magnitude of scatter, non existence n 2 of trends Δyin residuals = y F( and ϑ ) the ; Snumber = Δof free ; parameters ; needed g. We i i i y w δs = 0 ᆴ i i should tailor the model so that g is iminimal. = 1 n n ᆴ F( ϑ, a) (, ) Aptly selected Δ free parameters i 0; are then Δ results ᄊF ᆴ ϑ t b of the i y w = y phenomenological w = 0. i i i i modelling. i = 1 LSM technique ᄊa j then enables i = 1 reliable ᄊϑ i estimate ᄊb k of their uncertainty. 0

11 Eclipsing binaries The best template light curves can be obtained using EB physical models in PHOEBE or Wilson-Devinney codes. It was the case of very short period EB DD Com. We could use also the phenomenological model of light curve. We developed a bundle of EB LC models which are able to describe majority of real cases more than satisfactorily using minimum of free parameters. Example of model for close EB with zero eccentricity (minima at phases 0, 0.5), proximity effects, O Connel s effect and eclipses (even U shape minima) applicable for most of close binaries only 7 (!) free parameters. C f ( f ) exp F( ϑ) = m + a 1 1 exp + a 1 d d C + a cos( 4 πϑ) + a sin( 2 πϑ) sin( 6 πϑ) + sin( 10 πϑ) where f = ϑ round( ϑ) ; f = ϑ floo(r ϑ) ; 5,

12 Triple star AR Aurigae Detached eclipsing binary with HgMn primary shows well defined light-time effect. Times of minima were determined from original data by the method of phase shift of the observed and template light curves the scatter of them is 2.5 times smaller than scatter of timings taken from the literature. The causes of this difference are obvious: standard methods for times of minimum determination approximate the light curve in the minimum by a parabola, what does not correspond to the reality. These methods do not respect the fact that the VS is periodic its LC is known. O-C diagram suitable for astronomical textbooks

13 Instead conclusions - something for O-CD lovers O-C diagrams represent tradition in period analyses. At the top they are more illustrative then the method of direct modeling. Happily, we can create the classical O-C diagram using so called virtual O-C values for any subsets of observational data. In addition we can construct diagram depicturing changes of the instant period P(t) 1 2 ᆴ F ᆴ F ( O-C ) = P( t )Δ y ; W = w ; j j j j j ᄊϑ ᄊ ϑ n k ( O-C ) W ( O-C ) ϑ W j j j j j j = 1 j = 1 ( O-C ) k = ; δp =. n k k nk 2 W ϑ W j j j j = 1 j = 1 n k