New Analysis of the Light Time Eﬀect in TU Ursae Majoris · New Analysis of the Light Time...

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Astronomy & Astrophysics manuscript no. TU˙UMa˙07 c© ESO 2018November 1, 2018

New Analysis of the Light Time Effect in TU Ursae Majoris

Liska, J.1, Skarka, M.1, Mikulasek, Z.1, Zejda, M.1, & Chrastina, M.2

1 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlarska 2, CZ-611 37Brno, Czech Republic, e-mail: [email protected]

2 SNP 263/19, SK-916 01 Stara Tura, Slovakia

Received February ..., 2015; accepted March ..., 2015

ABSTRACT

Context. Recent statistical studies prove that the percentage of RR Lyrae pulsators living in binaries or multiple stellarsystems is considerably lower than it could be expected. This paradox can be better understood when detail analysis ofindividual candidates is performed. We give a detail investigation of the Light Time Effect of the most probable binarycandidate TU UMa which is complicated by secular variation of the pulsation period.Aims. This paper attempts to model possible Light Time Effect of TU UMa using a new code applied on formerlyavailable and newly determined maxima timings in order to confirm binarity and refine parameters of the orbit ofRRab component in binary system. The binary hypothesis is further tested also using radial velocity measurements.Methods. A new approach for determination of maxima timings based on template fitting which is also usable on sparseor scattered data is described. This approach was successfully applied on measurements from different sources. Fordetermination of orbital parameters of a double star TU UMa we developed a new code for analysis of Light TimeEffect involving also secular variation in pulsation period. Its usability was successfully tested on CL Aur – an eclipsingbinary with mass-transfer in a triple system showing similar changes in O-C diagram. Since orbital motion wouldcause systematic shifts in mean radial velocities (dominated by pulsations) we computed and compared our model withcenter-of-mass velocities. They were determined using high-quality templates of radial velocity curves of RRab stars.Results. Maxima timings adopted from the GEOS database (225) together with those newly determined from sky surveysand new measurements (149) were used for construction of O-C diagram spanning more than five proposed orbitalcycles. This data set is five times larger than data sets used by previous authors. Modelling of the O-C dependenceresulted in 23.36-year orbital period which translates to the lowest mass of the second component of about 0.33M⊙.Secular changes in pulsation period of TU UMa over the whole O-C diagram was satisfactorily approximated byparabolic trend with the rate of −2.33ms yr−1. For the confirmation of binarity we used radial velocity measurementsfrom nine independent sources. Although our results are convincing, additional long-term monitoring is necessary todefinitely confirm the binarity of TU UMa.

Key words. stars: variables: RR Lyrae – binaries: general – methods: data analysis – techniques: photometric – tech-niques: radial velocities – stars: individual: TU UMa

1. Introduction

A significant part of stars lives in double or multiple stel-lar systems. However, reviews of pulsating stars in multi-ple stellar systems (e.g. Szatmary 1990; Zhou 2010) clearlyshow the lack of stellar pairs with an RR Lyrae compo-nent. Current number of confirmed binaries comprising anRR Lyrae type pulsator can be counted on one hand.

Binarity of an object can be revealed in many differentways. For example, detection of eclipses, periodic radial ve-locity changes, or regular astrometric shifts in visual binarycan serve as a direct proof of binarity. A companion of a pe-riodic variable star can be detected also indirectly throughchanges in timings of light extrema, the so called LightTime Effect (hereafter LiTE). RR Lyrae stars are locatedgenerally in larger distance from the Earth, hence astromet-ric detection of binarity is highly unlike. Since the spectraof RR Lyrae stars are influenced by pulsations, discoveryof binary nature of stars through changes in position ofspectral lines is also difficult (e.g., Fernley & Barnes 1997;

Send offprint requests to: J. Liska,e-mail: [email protected]

Solano et al. 1997). Thus the most promising methods aredetection of eclipses and LiTE.

In the Large Magellanic Cloud three candidates for RRLyraes in eclipsing binaries were detected (Soszynski et al.2003). However, these objects were identified as opticalblends which consist from two objects, RR Lyrae star andeclipsing system (Soszynski et al. 2003; Prsa et al. 2008). Avery interesting object was identified by Pietrzynski et al.(2012) and subsequently studied by Smolec et al. (2013).This peculiar eclipsing system with orbital period of 15.24dcontains a component which mimics an RR Lyrae pulsator.The detailed study of Smolec et al. (2013) showed that thisobject, OGLE-BLG-RRLYR-02792, has a very low mass ofonly 0.26M⊙ which is too little for classical RR Lyrae stars.Also other physical characteristics indicated that this bi-nary component is rather a special object, which is a resultof evolution in a close binary, than a classical RR Lyrae star.Other object in the LMC, OGLE-LMC-RRLYR-03541, isrecently the best candidate for RR Lyr in eclipsing binarywith orbital period of 16.229 d (Soszynski et al. 2009).

Szatmary (1990) published a list of various types ofpulsating stars bounded in binary systems on the basis

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http://arxiv.org/abs/1502.03331v1

Liska et al.: Light Time Effect in TU Ursae Majoris

of visual identification of LiTE in their O-C diagrams.However, many of presented candidates are at least du-bious. Among others, this list contains TU UMa in whichthe LiTE with a 23-year long orbital period was proposedby Szeidl et al. (1986). The preliminary orbital parameterswere determined by Saha & White (1990). They also triedto verify the binarity using radial velocity measurements.Another examples of RR Lyrae stars in binaries were iden-tified through the analysis of their O-C diagrams very re-cently. Li & Qian (2014) found that FN Lyr and V894 Cygare probably in pairs with brown dwarves. Hajdu et al.(2015) found another 12 candidates among OGLE bulgeRRab variables. Nevertheless, these candidates should beconfirm by spectroscopy or different method. Catalogueof all kinds of binary systems with pulsating componentsfrom Zhou (2010, version December 2014) contains for RRLyrae class TU UMa, OGLE-BLG-RRLYR-02792 and sev-eral tens candidates without any closer information.

In this study, we present a new analysis of the LiTE inTU UMa which is based on much wider sample of O-C in comparison to the last paper about LiTE in TUUMa (Wade et al. 1999). A highly accurate photometricobservations, which cover two thirds of the proposed or-bital period, are newly available since Wade et al. (1999).In Section 2 we briefly discuss the history of observa-tion of TU UMa putting emphasis on its binary nature.In Section 3 the characteristics of the data sample usedare summarized. Except for data from various sourceswe utilized original measurements gathered in 2013-2014.Application of our LiTE procedure (described in Section 4and Appendix A) and modelling of the TU UMa data is pre-sented in Section 5. Other proofs for binarity are discussedin Section 6, and all results are summarized in Section 7.

2. History of observation of TU UMa

TU Ursae Majoris = AN 1.1929 = BD+30 2162 = HIP56088 (α = 11h29m48s.49, δ = +30◦04′02′′.4, J2000.0) isa pulsating RR Lyrae star of Bailey’s ab type. Accordingto Variable Star Index1 (Watson et al. 2006) its brightnessin V band varies in the range of 9.26–10.24mag (spectraltype A8–F8) with a period of about 0.558d. No signs of theBlazko effect have been reported for TU UMa up to now.

A variability of TU UMa was discovered byGuthnick & Prager (1929) on Babelsberg’s plates.Thereafter many authors studied the star using photoelec-tric photometry and spectroscopy. Detail description ofhistory of TU UMa research was performed by Szeidl et al.(1986). Only the most important information about theLiTE is briefly mentioned below, since about 170 articleswith keyword TU UMa are currently retrievable at theNASA ADS portal.

Payne-Gaposchkin (1939) was the first who noted cyclicvariations in maxima timings and proposed a 12400-day(34 years) long cycle. Important results were obtained bySzeidl et al. (1986), who mentioned secular period decreasewhich causes parabolic trend in O-C diagram, and a proba-ble 23-year (8400d) variations possibly caused by the bina-rity of the star. Saha & White (1990) detected systematicshifts in radial velocities (hereafter RVs) indicating bina-rity, but number of used RV measurements was very low.They modelled the LiTE for the first time, and determined

1 http://www.aavso.org/vsx/

orbital period of 7374.5d (20.19 yr). Their analysis showedthat the proposed stellar pair has an extremely eccentricorbit with e=0.970. They considered only constant pulsa-tion period in their model. An influence of neglecting thesecular changes of e, a sin i, and M2 sin3 i was tested byWade et al. (1992).

The LiTE with secular variation in the pulsation pe-riod was firstly solved by Kiss et al. (1995), who deter-mined more accurate orbital elements and determined or-bital period as 8800d (24.1 yr). Wade et al. (1999) collectedall available maxima timings and also RVs, and success-fully verified results from Saha & White (1990). They ob-tained five different groups of models of LiTE with respectto different subsets of maxima timings (all maxima with-out visual values, only photoelectric and CCD values, etc.).Orbital periods that they derived ranged from 20.26yrto 24.13 yr depending on particular data set and num-ber of fitting parameters (with/without parabolic trend).Subsequently, they used nine derived sets of orbital ele-ments for the reconstruction of orbital RV curve of pulsat-ing component, and for comparison with shifts in observedRVs. Since then, TU UMa has been neglected for about15 years from a viewpoint of study of LiTE. However, im-provements of quadratic ephemeris were performed by e.g.Arellano Ferro et al. (2013). A lot of high-accurate maximatimings (mainly CCD measurements) were published dur-ing this 15-year interval. Currently available data extendfive proposed orbital cycles.

3. Data sources

3.1. GEOS database

Since the GEOS RR Lyrae database2 (Groupe Europeend’Observations Stellaires, Le Borgne et al. 2007) is themost extended archive of times of maxima of RR Lyraestars, it was used as the main data source for our analysis.

Concerning GEOS values3, we paid special attentionto maxima timings based on data from sky surveys likeHipparcos (ESA 1997) or ROTSE = NSVS (Wozniak et al.2004). These timings are special in the meaning of methodof their determination, because in the most cases they aredetermined statistically based on the combination of manypoints which are often spread out over a few years.O-C val-ues determined from such data sets very often did not fol-low the general trend of O-C dependence4. Thus they wereomitted. However, we re-analysed the original data of thesesurveys (Sec. 3.3).

3.2. Our observations

As an extension to the GEOS data we also used ten newmaxima timings (in Appendix B) gathered by J. Liska in19 nights between December 2013 and June 2014. CCDphotometric measurements were performed using two tele-scopes – three nights with a 24-inch Newtonian telescope

2 http://rr-lyr.irap.omp.eu/dbrr/3 GEOS data marked as ‘pr. com.’ were not used.4 e.g. original value of maximum timing 2448500.0710 HJD

from Hipparcos satellite (Maintz 2005) has residual value O-C res = 0.0096 d based on model 2 in Sec. 5, but standard devi-ation of CCD measurements determined from the model is only0.0018 d.

2


Fig. 1. Differential magnitudes of TU UMa obtained with1-inch telescope folded with the pulsation period (blackdots) plotted together with the model of V -band obser-vations from (Liakos & Niarchos 2011).

(vby Stromgren filters) at Masaryk University Observatoryin Brno and 16 nights with a small 1-inch refractor(green filter with similar throughput as Johnson V filter,Liska & Liskova 2014) at private observatory in Brno. Inthe case of the small-aperture telescope, each 5 frames withexposure time of 30 s were combined to a single image toachieve better signal-to-noise ratio. The time resolution ofsuch combined frame is 150 s. Comparison star BD+30 2165was the same for both instruments, but control stars wereBD+30 2164 (for 24-inch telescope) and HD 99593 (for 1-inch telescope), respectively. Maxima timings were deter-mined via polynomial fitting or template fitting describedin Sec. 3.3. Monitoring with the small telescope resulted ina well covered phase light curve shown in Fig. 1. Exceptfor maxima timings determination, the observations wereimportant also for checking of possible eclipses (Sec. 6.2).

3.3. Other sources

We utilized high-cadence measurements from theSuperWASP project (Pollacco et al. 2006; Butters et al.2010) and Pi of the Sky project (e.g. Burd et al. 2004;Siudek et al. 2011) for determination of maximum timingsfrom the individual well-covered nights.

In addition, to maximally extend the O-C dataset wealso analysed data from other large sky surveys (Hipparcos,NSVS) and from the project DASCH (photometry fromscanned Harvard plates, e.g. Grindlay et al. 2009). Forthese, very sparse, but very extended data, maxima tim-ings were estimated using template fitting. The same pro-cess was applied on the data from the other sources.

Firstly, we chose the dataset with the best quality dataand with the best phase coverage. Since the data fromall surveys were of insufficient quality for constructionof the template light curve, V -band measurements fromLiakos & Niarchos (2011) were used. Subsequently we mod-

Table 1. Numbers of new maxima timings of TU UMadetermined from individual projects and from our observa-tions.

Hipparcos NSVS Pi of the Sky DASCH SuperWASP Our

3 4 5 57 64 10

elled the shape of the light curve in Matlab via non-linearleast-squares method with an n-order harmonic polynomial

m(t) = A0 +n∑

j=1

Aj cos

(

2 π jt−M0

Ppuls

+ φj

)

, (1)

where A0 is the zero level of brightness, Aj are ampli-tudes of the components, t is the time of observation inHeliocentric Julian Date (HJD), M0 is the zero epoch ofmaximal brightness, Ppuls is the pulsation period in days,and φj represent the phase shifts in radians. After someexperiments it was found that polynomial with n = 15 issufficient for good template model.

Nevertheless, using one template curve for various sur-veys brings some particularities. Firstly, the model had tobe scaled, because different surveys use different filters, andperiod and zero epoch had to be slightly refined for eachdataset. Subsequently, outliers were iteratively removed.

In case the data were sparse without well defined max-ima, it was necessary to divide the whole dataset to smallersubsamples containing typically about 30 points with thetime span from several days to hundreds of days. Datain these subsamples were then compared with the refinedtemplate light curve, and the time of maximum was de-termined. Logically, with this subsampling we were able toestimate only the mean time of maximum for the time in-terval of particular subset. However, in comparison withthe total time span of the O-C values, which cover severaldecades, this method is fully appropriate. All new maximafor TU UMa from surveys determined by polynomial ortemplate fitting (including uncertainties obtained directlyfrom least-squares method) are given in Appendix B. Thenumbers of all used maxima timings from particular surveysare in Table 1.

In addition, we determined 6 maxima timings from thedata which were omitted from the analysis in Boenigk(1958), Liakos & Niarchos (2011), Liu & Janes (1989) andPreston et al. (1961).

4. Modelling of LiTE

4.1. Light Time Effect

The existence of LiTE was suggested at the end of the 19thcentury in Algol system by Chandler (1888), while the first,detailed theoretical analysis of the problem was performedby Woltjer (1922). Irwin (1952a) solved important equa-tions for a part of the direct solution of LiTE and describedgraphical way of orbital elements determination.

Currently LiTE is usually solved very accurately ap-plying equations of motion in two-body system (equationsfrom Irwin (1952a) included direct solution of Kepler’sequation) or using numerical calculations of perturbed or-bit in multiple system. Nevertheless, the inverse part of

3


the calculations, in which searching for the best solu-tion is performed (minimization), still remains a problemfor discussion – various authors use various methods, e.g.damped differential corrections (Pribulla et al. 2000), least-squares method (Panchatsaram 1981), simplex (Levenberg-Marquart) method (Wade et al. 1999; Lee et al. 2010) orcombination of least-squares method and simplex (Zasche2008). Just for example, simplex method has several ver-sions, which differ in setting of initial parameters, sort-ing conditions, or in the size of corrections. Thus, ob-taining the same results, i.e. repeatability of process, isvery difficult, even impossible. To avoid this ambiguity,the inverse part of our code was constructed on the ba-sis of non-linear least-squares method described in (e.g.,Mikulasek & Zejda 2013) applied to modelling of LiTE (fordetails see Sec. 4.2 and Appendix A). This way was al-ready applied to analysis of LiTE in AR Aur system(Mikulasek et al. 2011) and is similar to the one used byVan Hamme & Wilson (2007) and Wilson & Van Hamme(2014).

4.2. Fitting procedure

The code that we used is written in Matlab. It consistsof several modules: loading measurements and setting ini-tial input parameters, direct solution of LiTE including op-tional parabolic trend, inverse minimisation method, and,finally, the selection of the best solution and calculation ofuncertainties of individual parameters through bootstrapresampling (Sec. 4.3).

A prediction of maxima timings Tcal calculated accord-ing to the relation

Tcal = M0 + Ppuls ×N +∆, (2)

contains linear ephemeris, as well as correction ∆ for LiTE.Parameter M0 is the zero epoch of pulsations in HJD, Ppuls

is the pulsation period in days,N is the number of pulsationcycle from M0.

The correction ∆ for LiTE includes calculation of theorbit of a pulsating star around the center of mass of abinary in relative units, and can be expressed by equationadopted from Irwin (1952a)

∆ = A

[

(1− e2)sin(ν + ω)

1 + e cos ν+ e sinω

]

, (3)

where e is the numerical eccentricity, ω is the argument ofperiastron in degrees, A

.= a1 sin i/173.145 is the projec-

tion of semi-major axis of primary component a1 in lightdays5 according to the inclination of the orbit i, and ν isthe true anomaly. The eccentric anomaly E, which is neces-sary for determination of the true anomaly ν, is solved fromKepler’s equation iteratively using Newton’s method with agiven precision higher than 1× 10−9 arcsec. Kepler’s equa-tion requires mean anomaly M , which is determined fromthe orbital period Porbit in days and the time of periastronpassage T0 in HJD.

Optional, more complex model, which includes aparabolic trend in O-C (e.g. Zhu et al. 2012), uses mod-ified equation (2) in a form of

Tcal = M0 + Ppuls ×N +1

2Ppuls Ppuls ×N2 +∆, (4)

5 semi-amplitude of LiTE changes in O-C diagram is thenALiTE = A

√1− e2 cos2 ω.

where parameter Ppuls = dPpuls/dt is the relative rateof changes of the pulsation period in [d d−1]. For easycomparison with other RR Lyrae stars, we used prescrip-tions from Le Borgne et al. (2007). Their parameter a3 =

1/2Ppuls Ppuls is the rate of period changes per one cycle in[d cycle−1], and the rate of period changes β in [ms d−1] isβ = 6.31152× 1010 a3/Ppuls or β = 0.07305× 1010 a3/Ppuls

in [dMyr−1]6.The first step of the non-linear least-squares method

is the linearisation of the non-linear model function(eq. 2 or 4) by Taylor decomposition of the first order (seeMikulasek et al. 2006; Mikulasek & Graf 2011)

Tcal∼= Tcal(T,b0) +

g∑

j=1

∆bj∂Tcal(T,b)

∂bj, (5)

where bj are individual free parameters in vector b. Vectorb0 contains initial estimates of parameters, ∆bj are theircorrections, g is a number of free parameters (the length ofmatrix b).

After linearisation, the problem can be solved in thesame way as in the linear least-squares method, but withseveral necessary iterations to obtain a precise solution.Initial parameters are in our code quasi-randomly gener-ated many times from large interval with limits defined byuser. The derivatives are solved analytically. For more de-tails see Appendix A.

Parameter χ2(bk) or its normalised value χ2R(bk) =

χ2(bk)/(n− g), where n is a number of measurements, wasused as an indicator of the quality of the k-fit.

Since many maxima timings from the GEOS databaseare given without errors or they are often questionable, analternative approach for weights determination was applied.The dataset was divided in several groups according to thetype of observations (photographic, visual, photoelectric,CCD, DSLR), and weights were assigned to each of thegroups with respect to the dispersion of points around themodel. Values of weights were improved iteratively. Groupwith less than five points (DSLR) were merged with anothergroup with similar data quality to avoid unrealistic weightassignment (CCD+DSLR). During fitting process outliersdiffering more than 5 σ from the model were rejected. Visualsupervision was applied in all steps of the analysis.

The LiTE fitting process does not allow an estimationof masses of both stars, but only a mass function f(M)

f(M) =(M2 sin i)3

(M1 +M2)2=

4 π2

G

(a1 sin i)3

P 2orbit

, (6)

where M1, M2 are masses of the components, i is the in-clination angle of the orbit, Porbit is the orbital period,and a1 is the semi-major axis of the primary component.Based on the studies of Fernley (1993) and Skarka (2014)we adopted the value for the mass of RR Lyrae componentas M1 = 0.55M⊙, and inclination angle was set to i = 90 ◦

(sin i = 1). This allows computation of the lowest mass ofthe second component solving cubic equation

M32 − f(M)M2

2 − 2 f(M)M1M2 − f(M)M21 = 0. (7)

In this work we expect that variation in O-C diagram ofTU UMa can be well described using eq. 4 and the other

6 Le Borgne et al. (2007) probably used a length of the year366 d therefore their constant 0.0732 × 1010 is a little different.

4


possible secular variation of pulsation period could be ne-glected (it has low amplitude or it appears in longer timescale).

4.3. Bootstrap-resampling

The (non-linear) least-squares method itself gives an es-timation of uncertainty of all fitted parameters, but themethod is extremely sensitive to data characteristics. Aslight change of the dataset by adding one single mea-surement can cause significant difference of new parametersfrom the previous solution. Thus we decided to use a sta-tistical approach represented by bootstrap-resampling forestimation of the errors.

Parameters from the best solution were used as initialvalues for fitting of a new dataset, whose points were ran-domly selected from the original dataset. This procedurewas repeated 5000 times. From the scattering of individualparameters of these five thousands solutions their uncer-tainties were estimated. Errors in Tables 2, 3 correspond to1σ.

4.4. Test object CL Aurigae: eclipsing binary with probableLiTE and mass-transfer

The code was, among others, tested on a well known de-tached binary system, CL Aur. This eclipsing binary waschosen intentionally, because it shows LiTE and a secularperiod change. In addition, CL Aur was studied in similarway three times during the last 15 years (Wolf et al. 1999,2007; Lee et al. 2010).

The times of minima of CL Aur taken from O-C gate-way database7 (Paschke & Brat 2006) were used to con-struct O-C diagram and to determine parameters throughthe methods described above8. Our best model (Fig. 2) de-scribes O-C variations very well in the most recent part(precise CCD observations). The old part of O-C diagramwith visual and photographic measurements is highly scat-tered, but these measurements were also taken into ac-count during fitting process by assigning them with a lowerweight (model weights for different observation methodswere found in ratio pg:vis:ccd 1:11.5:403)9. We can con-clude that our results are comparable with previous re-sults (Table 2). This example, as well as additional testing,showed that our code works well, and is suitable for analysisof RR Lyraes with suspected LiTE. The code was alreadysuccessfully used for modelling LiTE in V2294 Cyg (Liska2014).

5. Light Time Effect in TU UMa and analysis of

O-C diagram

Cyclic changes in O-C diagram of TU UMa are wellknown for a long time, and were analysed in detail bySaha & White (1990), Kiss et al. (1995), and Wade et al.

7 http://var.astro.cz/ocgate/8 The model was calculated for the half value of the period.

Subsequently, results were corrected for this effect.9 Compare: Wolf et al. (2007) visually distinguished quality of

data by weights in each category 0, 1, 2 for pg, 0, 1, 2 for visual,and 5, 10, 20 for CCD observations, respectively.

10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000−0.05

0.00

0.05

0.10

0.15

0.20

HJD − 2400000 [d]

O−

C [d

]

Fig. 2. O-C diagram of the testing eclipsing binary CLAurigae (double star without RR Lyrae component) withLiTE and parabolic trend (black circles). Model of changes(red line) is based on our parameters from Table 2.

Table 2. Our determined parameters of the testing objectCL Aur (right) together with results from previous studies(left).

Study Wolf et al. (2007) Lee et al. (2010) our model

P [d] 1.24437505(18) 1.24437498(17) 1.24437488+16−12

M0 [HJD] 2450097.2712(5) 2450097.27082(46) 2450097.27155+61−66

10−10P4.05(6)∗ 3.92(55)∗ 3.76+28

−22[d d−1]

10−10 a32.52(4) 2.44(34) 2.34+17

−14[d cycle−1]

β [ms yr−1] 12.8(2)∗ 12.4(1.7)∗ 11.86+89−69

β [dMyr−1] 0.148(2) 0.143(20) 0.137+10−8

P3 [yr] 21.7(2) 21.63(14) 21.61+19−18

T0 [HJD] 2443880(80) 2444072(56) 2444020+140−190

e 0.32(2) 0.337(53) 0.271+47−34

ω [◦] 209.2(1.2) 218.9(2.7) 218.4+6.0−9.2

A [light day] 0.0144(12)∗ 0.01378(72)∗ 0.01388+31−22

a12 sin i [au] 2.49(22)∗ 2.38(12) 2.404+54−38

f(M3) [M⊙] 0.034 0.0290(15) 0.0297+19−14

K12 [km s−1] − − 3.44+10−6

χ2R − − 1.04(10)

Nmin 144 198 203

Notes. (∗) Parameter was calculated using values from originalstudy.

(1999). The parameters determined by these authors aregiven in Table 3.

Our dataset described in Sec. 3 is much denser and moreextended than in previous studies (we used 424 maximatimings, in contrast to Wade et al. (1999) who used only83). The used O-C values spans 115 years mainly due tomeasurements recorded on the Harvard plates (provided bythe project DASCH) in the first half of the 20th century.Because the complete dataset is not homogeneous withuncomparably better coverage over the last two decades(CCD measurements), we analysed LiTE in TU UMa in

5


10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000−0.08

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

HJD − 2400000 [d]

O−

C [d

]

30000 35000 40000 45000 50000 55000 60000−0.025

−0.020

−0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

HJD − 2400000 [d]

O−

C [d

]

Fig. 3. O-C diagram of TU UMa. Period decrease man-ifested by the parabolic trend (dotted line) is obvious.Cyclical changes due to an orbital motion are also clearlyremarkable. Our model of LiTE is represented by the solidred line. The top panel shows the model 1 with all availabledata, while the plot in the bottom panel shows the situa-tion with only photoelectric, CCD and DSLR observations(model 2).

two ways. Model 1 is based on the whole dataset10, whilemodel 2 describes only photoelectric, CCD and DSLR ob-servations11. Since TU UMa experiences secular period de-crease (in Fig. 3 represented by the parabolic dashed curve)with the rate about−2.9×10−11 days per cycle (Wade et al.1999), we used the complex form of the model (eq. 4).

Logically, model 1 based on the whole data set (cover-ing 5 orbital periods) gives more precise, but slightly dif-ferent results than model 2, which spans only two of the23-year orbital cycles. Nevertheless, high-accurate photo-electric and CCD measurements cover the whole orbitalcycle very well (see Fig. 4). Due to the shorter time base,our second model has, for example, lower value of secularperiod evolution. This is carried at the expense of increasingeccentricity (0.63 and 0.67 for the model 1 and 2, respec-tively).

In comparison with orbital elements from previous stud-ies given in Table 3, our results are of better confidence

10 Weights for model 1 were found in ratiovis : pg : pe : CCD+DSLR 1.0 : 3.0 : 40.8 : 60.5, uncertainties0.0143 d, 0.0083 d, 0.0022 d, 0.0018 d.11 Weights for model 2 were in ratio pe : CCD+DSLR1.00 : 1.22, uncertainties 0.0020 d, 0.0018 d.

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2−0.020

−0.015

−0.010

−0.005

0.000

0.005

0.010

0.015

0.020

Orbital phase

O−

C [d

]

Fig. 4. O-C diagram of TU UMa constructed only fromphotoelectric, CCD and DSLR measurements, after sub-traction parabolic trend, and phased with the orbital periodbased on model 2.

due to larger and better dataset. Our values differ mainlyin eccentricity and distance between components, as wellas in the mass function. All these values were foundto be significantly lower than those from Saha & White(1990), Kiss et al. (1995) or Wade et al. (1999). Valuesfrom Saha & White (1990) differ more due to neglecting pe-riod decreasing and due to the missing expression (e sinω)in their eq. 3 which should be similar to our eq. 4.2.

From our results it seems that TU UMa is very likely amember of a well-detached system with a dwarf componentwith the minimal mass of only 0.33 M⊙. Since no signs ofthe companion are observed in the light of TU UMa, it isprobably a late-type main sequence dwarf star. However,we cannot exclude the possibility that it is a white dwarfor a neutron star, since we do not know the inclination.

Explanation of the non-zero eccentricity in this old sys-tem, which has had definitely enough time for circulariza-tion, remains an open question. The most probable candi-date for the explanation is some explosive process in thesystem (Saha & White 1990). This scenario would prefer adegenerate remnant as a second component. Another sce-nario, no matter how unlike, is that the second body wasattracted and the circularization is still in progress.

Except for the LiTE, which is the most remarkable fea-ture of the O-C diagram, also changes represented by theparabolic trend are apparent. The progression of the de-pendence suggest secular shortening of the pulsation pe-riod of TU UMa which is almost certainly an evolution-ary effect, because the mass transfer, which is responsi-ble for period changes in close binaries, can be excludeddue to a very wide orbit of TU UMa. In addition, valueβ = Ppuls∼−2.3ms yr−1=−0.027dMyr−1 can correspondto blueward evolution of the RR Lyrae component, nev-ertheless e.g. Le Borgne et al. (2007) give higher medianvalue β = −0.20dMyr−1 for their sample of 21 stars withsignificant period decreasing.

After subtraction of our model 1 from the wholedataset (Fig. 5), several photographic measurements (inrange between JD 2426000 and 2432000) are more devi-ated than other values with systematic shift about 15 min-utes (0.01 d). It could indicate that TU UMa could undergomore complex period changes than only LiTE and parabolictrend. Some of possible explanations such as cubic trend

6


Table 3. Parameters determined previously and our results for the system TU UMa. Mass limit of the second body wasestimated from the mass function f(M), inclination of orbit (i = 90◦) and mass of RR Lyrae star M1 = 0.55M⊙ adoptedfrom Fernley (1993) and Skarka (2014). Our parameters (right part of the table) were calculated from all maxima timingsfor model 1 and only from photoelectric, CCD and DSLR measurements for model 2.

Study Saha & White (1990) Kiss et al. (1995) Wade et al. (1999)C model 1 model 2

Ppuls [d] 0.5576581097 0.5576581097 0.55765817(29)D 0.557657610+17−16 0.557657483+20

−24

M0 [HJD] 2425760.4364 2425760.4364 2425760.464(5)D 2442831.48678+44−50 2442831.48625+42

−43

Ppuls× −31.48∗ −10.4(7)∗ −7.37+25

−24 −4.59+52−4410−11[d d−1]

a3=1/2PpulsPpuls× −8.78∗ −2.9(2) −2.056+70

−66 −1.28+15−1210−11 [d cycle−1]

β= Ppuls [ms yr−1] × −9.934∗ −3.3(2)∗ −2.327+79−74 −1.45+16

−14

β= Ppuls [dMyr−1] × −0.11498∗ −0.038(3)∗ −0.02693+91−86 −0.0168+19

−16

Porbit [yr] 20.19∗ 24.1(3)∗ 23.27(24)∗ 23.363+79−63 23.263+67

−75

T0 [HJD] 2425000∗ 2447200(50) 2421585(207) 2447090+41−46 2447138+44

−37

e 0.970 0.90(5) 0.74(10) 0.631+28−28 0.673+26

−24

ω [◦] 196.1∗ 178(3) 183(5)∗ 181.4+2.0−2.1 184.8+2.0

−2.0

A [light day] 0.100 0.023(5)∗ 0.0203(35)∗ 0.01675+53−47 0.01723+59

−51

a1 sin i [au] 17.3∗ 4.0(7)∗ 3.52(61) 2.900+92−81 2.98+10

−9

f(M) [M⊙] − 0.11(1) 0.080 0.0447+43−37 0.0491+51

−42

M2,min∗ [M⊙] − 0.17 − 0.325+13

−13 0.338+15−14

K1 [km s−1] − 11.4(5) 6.6 4.77+29−26 5.17+0.35

−0.29

χ2R − − − 1.060(69) 1.025(90)

Nmax ∼ 43 ∼ 42 67 424 255

Notes. (∗) Parameter was calculated using values from original study, (C) their approach C was selected, (D) pulsation elements are knownonly from their approach D.

10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000−0.04

−0.03

−0.02

−0.01

0.00

0.01

0.02

0.03

HJD − 2400000 [d]

O−

C [d

]

Fig. 5. Residual O-C diagram of TU UMa (for greater clar-ity without visual observations) after subtraction of the 1stmodel of LiTE. Jump in general trend of O-C in range fromJD 2426000 to 2432000 could be an indication of the morecomplex period changes.

or additional LiTE could describe this variation. However,the residuals are very scattered and also some instrumen-tal artefacts can play role. Thus our explanations are notconclusive.

6. Other proofs for binarity

Since LiTE is only indirect manifestation of the binarity, itis necessary to prove it in other way. The analysis of themean radial velocities can be considered as the most valu-able test. Except for this method, other possible approaches

for confirmation of binarity of TU UMa are discussed in thenext sections.

6.1. Radial velocities

Known orbital parameters from the analysis of LiTE allowus to predict, but also reconstruct, the RV curve from thepast. The binarity can then be proved via comparison ofthe model for the orbital RV curve and spectroscopicallydetermined center-of-mass RV. For TU UMa such analy-sis was firstly performed by Saha & White (1990), and fewyears later also by Wade et al. (1999). They noticed sys-tematic shifts in RV determined in different times. Theirpredictions correlated with measurements fairly well.

We scanned literature for RV measurements and foundnine sources (Table 4). Unfortunately, the last availabledata with RVs was published in 1997. Saha & White (1990)used only 3 sources of RVs; Wade et al. (1999) do not givetheir values. In addition, both authors ignored RV mea-surements from Preston & Paczynski (1964). Other authorsgive slightly different values determined from the samedataset (Table 5, column 2) and without mean time of ob-servation. Therefore we decided to re-analyse all availableRV measurements.

Determination of the center of mass RV for a binarywith pulsating star is more complicated than for a bi-nary with non-variable stars (pulsations are often domi-nant source of RV changes). Another inconveniences areconnected with RV based on different type of spectral lines(e.g. Balmer or metallic lines). They are formed in differ-ent depths and therefore they have different shape, am-plitude, and zero points (already shown e.g. by Sanford1949; Oke et al. 1962). Since available RV measurements

7


Table 4. Sources of radial velocity measurements for TUUMa, S is a source number, NRV is a number or radialvelocity measurements.

S Publication NRV Lines

1 Abt (1970) 1 Unknown

2 Barnes et al. (1988) 74 Metallic

3 Fernley & Barnes (1997) 3? OI triplet, (Hα)

4 Layden (1994) 5 Hydrogen, CaII K

5 Liu & Janes (1989) 60 Metallic

6aPreston et al. (1961)

4 Metallic

6b 21 Hγ , Hδ, H8−11, Ca II K

7aPreston & Paczynski (1964)

12 Metallic

7b 8+7 Hydrogen

8 Saha & White (1990) 32 Metallic

9 Solano et al. (1997) 3? Metallic, (Hγ )

are based on various lines it was necessary to unify them.This was done using high-accurate normalised templatecurves from Sesar (2012). Firstly we modelled these tem-plate curves with n-order harmonic polynomial. The ob-served RV curve for particular spectral line was then com-pared with polynomial image of the template, and the am-plitude and the central value of RV curve was determinedby the least-squares method simultaneously for all datasets.Before this step, measurements were time-corrected for bi-nary orbit and period shortening (based on model 1) andseveral of the datasets were divided into smaller groups toobtain time resolution about 1-year (see Table 5 with de-termined mean RVs for given epochs corresponding to themean value of observation time). We did not found orig-inal RV measurements for two studies (Fernley & Barnes1997; Solano et al. 1997) and thus we only adopted theirmean RV values and estimated mean time of observationfrom information in their papers. Finally, the mean centerof mass RV values were then compared with the RV modelresulting from our analysis of LiTE (Fig. 6). It is seen thatpoints roughly follow the model RV curve.

An alternative test for binarity using RV curves can beperformed by the comparison of observed RV curves (thetop panel of Fig. 7) and those in which the orbital RV curvefrom the model is subtracted (the bottom panel of Fig. 7).In this figure RV measurements are phased according to thepulsation period12. Apparently, the phased RV curve withcorrected velocities is significantly less vertically scatteredthan without the correction. The residual scatter in thebottom panel results from different metallic lines that theRVs were based on.

Both tests clearly show that TU UMa is very likelybounded in a binary system.

12 The stitching in phase was possible only with taking theLiTE and secular period change into account. Otherwise thecurve would be scattered horizontally.

Table 5. Determined values of center-of-mass velocities forTU UMa based on different measurements and templatesfrom Sesar (2012). Mean values published in different pub-lications are present for comparison, S is a source numberof original RV measurements.

S RVpub Tmid RVour errRVour

[km s−1] [HJD] [km s−1] [km s−1]

1 104(35)La 2426076 104La 35La

2 90(2)F , 90(2)So 2443563 95 3

2443941 95 2

2444218 90 1

2444948 88 5

3 101(3)F , 101(5)So 2449520 101F 3F

4 75(17)La , 75(17)F 2447975 78 13

5 84.2Li, 84Sa, 84(1)La, 2446843 84 1

84(1)F , 84(2)So 2447130 85 3

6a – 2436979 93 1

6b 92(1)P , 87H , 2436647 93 2

93(3)Sa, 92(1)F 2436979 94 4

7a – 2438039 94 2

7b – 2438039 95 2

8 77Sa, 77(2)F , 77(2)So 2446894 76 1

9 96(3)So 2449596 96So 3So

Notes. Value was adopted from: (F )Fernley & Barnes (1997),(H)Hemenway (1975), (La)Layden (1994), (Li)Liu & Janes

(1990), (P )Preston et al. (1961), (Sa)Saha & White (1990),(So)Solano et al. (1997).

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

−20

−15

−10

−5

0

5

10

15

20

Orbital phase

RV

[km

/s]

model 1model 2obs RV − 89 km/s

Fig. 6. Models of variations in RV caused by orbit of pul-sating component around mass-centre of the binary system(red and blue lines) and center-of-mass velocities deter-mined for each dataset of RV measurements using templatefitting or adopted from literature. The visually estimatedcorrection −89 kms−1 for systematic velocity mass-centreof the system from Sun (γ-velocity) was applied.

6.2. Eclipses

The detection of eclipses in the light curve (in appropriatephase of the orbit) would be a strong proof for binarityof TU UMa. Among eclipsing binary stars, several thirdcomponents, known only from the LiTE, were confirmed by

8


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

40

50

60

70

80

90

100

110

120

130

Corrected phase

RV

[km

/s]

256a7a8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

40

50

60

70

80

90

100

110

120

130

Corrected phase

Cor

rect

ed R

V [k

m/s

]

256a7a8

Fig. 7. Radial velocity curves from metallic lines of TUUMa from different publications phased with pulsation pe-riod corrected for LiTE and secular period changes. Notcorrected observed radial velocities (top) and corrected val-ues after subtracting changes in RV caused by orbital mo-tion based on our model 1 (bottom). RV values correctedfor binary orbit are evidently less scattered than not cor-rected RVs.

detection of additional eclipses (e.g. in the Kepler project,Slawson et al. 2011). The chance to catch an eclipse in TUUMa is very low, because expected orbital period of thebinary system is very long, inclination angle of the orbit isunknown and radius of secondary component is probablymuch smaller than for the pulsating star. Both our modelsof LiTE allow us to estimate the time of possible eclipse(winter 2013 or summer 2014), but difference between bothpredictions is too large. However, we attempted to detectproposed eclipse.

Observations with the small telescope described inSec. 3.2 were dedicated for this purpose. Unfortunately, ourmeasurements were insufficient for reliable decision abouteclipses. Due to weather conditions, limited object visibilityand other influences we observed only in 19 nights, whichcould be hardly sufficient regarding the imprecise eclipseprediction. At least we can conclude that no sign of eclipsewith amplitude higher than 0.07 mag was detected in ourdata (see Fig. 8).

7. Summary and conclusions

In this work, new analysis of the probable LiTE in TU UMawas performed. We used published maxima timings from

Fig. 8. Residuum of the light curve of TU UMa after sub-traction of harmonic polynomial model. No signs of eclipsewith amplitude higher than 0.07 mag was detected in greenband.

the GEOS database (275 values), and added values ofmaxima from our photometric observations and from theSuperWASP and Pi of the Sky surveys. We applied tem-plate fitting method to determine maxima from thesemeasurements and also from sparse data from projectsHipparcos, NSVS and DASCH. Altogether we analysed 424measurements of maxima timings, which is about 5 timeslarger dataset than the one from the last study of TU UMaby (Wade et al. 1999). This large and well covered datasetallowed us to determine a quadratic ephemeris of the pulsa-tions and orbital elements of the binary system with muchbetter accuracy than in previous studies (Table 3). All anal-yses were performed with a new code written in Matlabwhich uses bootstrap method for error estimations. We cal-culated two models: model 1 which describes the wholedataset, and model 2 which describes only high-accuratephotoelectric, CCD and DSLR maxima. The second modelis based on data with significantly shorter time span thanfor the model 1.

The second model gives a smaller value of period-decrease rate (β = Ppuls ∼ −1.5ms yr−1), which causesthe eccentricity to become higher (e ∼ 0.67) than in thefirst model (β ∼ −2.3ms yr−1, e ∼ 0.63). For comparison,Arellano Ferro et al. (2013) give β = −1.3ms yr−1 with-out LiTE fitting. Nevertheless, both our models have lowervalue of eccentricity, semi-major axis of pulsating com-ponent a1 sin i (2.9 au or 3.0 au) and minimal-mass limitof secondary component (0.33M⊙ or 0.34M⊙) than inprevious works. Our values of orbital period (23.4 yr or23.3 yr), argument of periastron ω (181◦ or 185◦), and semi-amplitude of radial velocity variations of pulsating star K1

(4.8 km s−1 or 5.2 km s−1) are comparable with values de-termined by previous authors.

The binary nature was tested in several ways. Firstly,our models of the orbit gave predictions of possible eclipses.Despite that the prediction was highly inaccurate and thatthe eclipse is highly unlike (wide orbit, unknown inclinationand other important parameters) we attempted to detectthem. Unfortunately without success. Binarity should man-ifest itself in cyclic changes of the mean radial velocity. We

9


adopted RV measurements from nine independent sourcesand corrected their values according to our model by sub-traction of LiTE and secular changes. When observed RVcurve was phased with the pulsation period, we got typicalRV curve for RR Lyrae which was scattered. The scattersignificantly dropped down when our model was applied.

We also determined central values of radial velocity foreach RV dataset using pulsation templates for differentspectral lines from Sesar (2012). We compared these val-ues with our model of orbital RV variations based on or-bital parameters known from LiTE. Evident correlation isapparent (Fig. 6).

The two successful proofs are important towards confir-mation of the binarity of TU UMa. However, only long termspectroscopic measurements covering the whole orbital cy-cle could unambiguously confirm that TU UMa is really amember of a binary system.

Acknowledgements. The DASCH project at Harvard is grateful forpartial support from NSF grants AST-0407380, AST-0909073, andAST-1313370. This paper makes use of data from the DR1 of theWASP data (Butters et al. 2010) as provided by the WASP consor-tium, and the computing and storage facilities at the CERIT ScientificCloud, reg. no. CZ.1.05/3.2.00/08.0144 which is operated by MasarykUniversity, Czech Republic. This research has made use of NASA’sAstrophysics Data System. Work on the paper have been supportedby MUNI/A/1110/2014 and LH14300.

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Appendix A: Application the non-linear

least-squares method on calculation of LiTE

Let us assume a group of n maxima timings given inHeliocentric Julian Date13 (HJD) determined from obser-vations. Times are inserted in column vector y with sizen × 1. Each of time of observed maximum Tl has corre-sponding uncertainty σl. It was supposed that quality ofl-measurement can be quantified by l-weight using relationwl = σ−2

l and these weights are inserted in column vectorw. For correct calculation, weights are normalised (averagevalue of weights is w = 1) and are inserted in a squarematrix W = diag(w).

13 Times of maxima in Baryocentric Julian Date should beused, nevertheless baryocentric correction is under the accuracyof maxima times from database GEOS which contains times inHJD valid to 4 decimal place. Also the accuracy of the deter-mined maxima are mostly worse – especially for photographicor visual measurements.

10

http://arxiv.org/abs/1502.01318





In the next step, an equation for model functionTcal(T,b) with unknown parameters in vector b is selected.Changes in position times of maxima for pulsating stardue to LiTE (periodic changes in O-C diagram) can beexpressed by equation

Tcal(T,b) = M0 + Ppuls ×N +∆, (A.1)

where M0 is the zero epoch of pulsation in HJD, Ppuls isthe pulsation period in days, N is the number of pulsationcycle from M0 and parameter ∆ is the correction for LiTE.The integer number of pulsation cycle (epoch) N is given

N = round

(

T −M0

Ppuls

)

. (A.2)

Optional, more complex model, which includes parabolictrend in O-C diagram (e.g. Zhu et al. 2012), uses modifiedequation (A.1) in a form of

Tcal(T,b) = M0 + Ppuls ×N +1

2Ppuls Ppuls ×N

2 +∆, (A.3)

where parameter Ppuls = dPpuls/dt is the relative rate ofchanges of pulsation period.

The correction ∆ for LiTE includes calculation of theorbit of pulsating star around mass-centre of binary in rel-ative units and is given by equation adopted from (Irwin1952a)

∆ = A

[

(1− e2)

sin(ν + ω)

1 + e cos ν+ e sinω

]

, (A.4)

where e is numerical eccentricity, ν true anomaly, ω argu-ment of periastron in degrees and A constant in light days,which compares the shift in a radial position to the timedelay caused by constant speed of light. True anomaly ν iscalculated from equation

tanν

2=

√

1 + e

1− etan

E

2, (A.5)

and eccentric anomaly E is determined in our code itera-tively by Newton’s method from Kepler’s equation

E = M + e sinE. (A.6)

Mean anomaly M is in form

M =2 π (T − T0)

Porbit

. (A.7)

where orbital period Porbit is in days, time of periastronpassage T0 in HJD.

The constant A is the projection of semi-major axis ofthe pulsating component a1 in the unit light day

A =a1 sin i au

86400 c.=

a1 sin i

173.145, (A.8)

where i is the inclination angle of the orbit in degrees, au islength of the astronomical unit in metres, c is speed of thelight in vacuum in m s−1. Semi-amplitude of LiTE changesin O-C diagram in days is then

ALiTE = A√

1− e2 cos2 ω. (A.9)

Subsequently, observed values of time of maximum canbe compared with the ones from the model obtained fromeq. A.1 and A.3, respectively. Their difference for the givenset of parameters is equal to

δTl = Tl − Tcal,l(Tl,b). (A.10)

The least-squares method (hereafter LSM) described in de-tail in (Mikulasek & Zejda 2013) says that the best modelhas the least sum of squares of residua between observationand model. Modified form of LSM used weighted form

δTmod,l =δTl

σl

=Tl − Tcal,l(Tl,b)

σl

, (A.11)

and than the sum is

χ2(b) =

n∑

l=1

δT2mod,l =

n∑

l=1

[

Tl − Tcal,l(Tl,b)

σl

]2

=

=

n∑

l=1

[

δT2l wl

]

. (A.12)

Its normalised value χ2R(bk) = χ2(bk)/(n− g), where n is

number of measurements and g is number of free parame-ters (the length of matrix b), was used as an indicator ofthe quality of the k-fit.

The first step of non-linear LSM is linearisation of thenon-linear model function (eq. A.1 or A.3) by Taylor de-composition of the first order (see Mikulasek et al. 2006;Mikulasek & Graf 2011)

Tcal∼= Tcal(T,b0) +

g∑

j=1

∆bj∂Tcal(T,b)

∂bj, (A.13)

where bj are individual free parameters in vector b, vectorb0 contains initial estimates of parameters, ∆bj are theircorrections.

Important equations for solution of LSM by matricesare

U = XTWy, V = X

TWX, H = V

−1 = (XTWX)−1 (A.14)

and matrix with derivatives is in form

X =

[

∂Tcal

∂M0,∂Tcal

∂Ppuls

,∂Tcal

∂T0,

∂Tcal

∂Porbit

,∂Tcal

∂A,

,∂Tcal

∂ω,∂Tcal

∂e

]

, (A.15)

where individual derivatives are also presented

∂Tcal

∂M0= 1,

∂Tcal

∂T0=

∂∆

∂ν

∂ν

∂E

∂E

∂M

∂M

∂T0, (A.16)

∂Tcal

∂Ppuls

= N,∂Tcal

∂Porbit

=∂∆

∂ν

∂ν

∂E

∂E

∂M

∂M

∂Porbit

, (A.17)

∂Tcal

∂A=

(1− e2) sin(ν + ω)

1 + e cos ν+ e sinω, (A.18)

∂Tcal

∂ω= A

[

(1− e2) cos(ν + ω)

1 + e cos ν+ e cosω

]

, (A.19)

∂Tcal

∂e=

∂∆

∂e+

∂∆

∂ν

∂ν

∂e+

∂∆

∂ν

∂ν

∂E

∂E

∂e. (A.20)

Other necessary derivatives are

∂∆

∂ν=

A (1− e2)

1 + e cos ν

[

cos(ν + ω) +e sin(ν + ω) sin ν

1 + e cos ν

]

, (A.21)

∂∆

∂e= A

{

sinω −sin(ν + ω) [2 e+ (1 + e2) cos ν]

(1 + e cos ν)2

}

, (A.22)

11


∂ν

∂E=

√

1 + e

1− e

cosν

2

cosE

2

2

,∂ν

∂e=

sin ν

1− e2, (A.23)

∂E

∂M=

1

1− e cosE,

∂E

∂e=

sinE

1− e cosE, (A.24)

∂M

∂Porbit

=−2π (T − T0)

P 2orbit

,∂M

∂T0=

−2π

Porbit

. (A.25)

Matrix X will be expanded about 1 additional member forcalculation of parabolic trend according to eq. A.3

X =

[

∂Tcal

∂M0,∂Tcal

∂Ppuls

,∂Tcal

∂T0,

∂Tcal

∂Porbit

,∂Tcal

∂A,

,∂Tcal

∂ω,∂Tcal

∂e,∂Tcal

∂Ppuls

]

, (A.26)

where two of X members are in form

∂Tcal

∂Ppuls

= N +1

2Ppuls ×N

2,

∂Tcal

∂Ppuls

=1

2Ppuls ×N

2. (A.27)

Determined parameters allow calculation of the radialvelocity (RV) changes caused by the secondary component(e.g. Irwin 1952b)

RV1 = γ +K1 [cos(ν + ω) + e cosω] , (A.28)

where γ is systematic velocity mass-centre of the binarysystem from Sun in km s−1 (γ-velocity) and K1 is semi-amplitude of RV changes in km s−1 given by equation

K1 =2π a1 sin i au

8.64 × 107 Porbit

√

(1− e2), (A.29)

where projection of semi-major axis a1 sin i is in au, con-stant au in meters, and orbital period Porbit in days.

12


Appendix B: New maxima timings for pulsating star TU UMa

Tmax [HJD] error [d] Project Method Tmax [HJD] error [d] Project Method Notes

2414783.4832 0.0020 DASCH pg 2432312.9088 0.0037 DASCH pg

2415424.2371 0.0042 DASCH pg 2433169.4834 0.0035 DASCH pg

2415767.7412 0.0033 DASCH pg 2433763.4027 0.0072 DASCH pg

2416164.8074 0.0024 DASCH pg 2434479.4402 0.0015 Boenigk (1958) pg



2416928.2510 0.0047 DASCH pg 2436654.8847 0.0003 Preston et al. (1961) pe V

2417300.2057 0.0065 DASCH pg 2442525.8986 0.0046 DASCH pg

2417698.9378 0.0031 DASCH pg 2443210.6868 0.0029 DASCH pg

2418264.4070 0.0020 DASCH pg 2445123.4726 0.0043 DASCH pg

2418445.0843 0.0020 DASCH pg 2445712.9160 0.0035 DASCH pg

2418776.3408 0.0028 DASCH pg 2446169.6364 0.0044 DASCH pg

2419413.7423 0.0062 DASCH pg 2446599.5863 0.0038 DASCH pg

2419497.4018 0.0025 DASCH pg 2447196.8232 0.0048 DASCH pg

2419905.0456 0.0027 DASCH pg 2447226.9383 0.0005 Liu & Janes (1989) pe V

2420307.6804 0.0035 DASCH pg 2447595.5451 0.0042 DASCH pg

2420609.3764 0.0040 DASCH pg 2448076.7991 0.0004 Hipparcos CCD Clear



2421954.9737 0.0017 DASCH pg 2451279.9925 0.0014 NSVS CCD Clear




2424225.7694 0.0023 DASCH pg 2453131.4272 0.0004 SuperWASP CCD CCD-103, WASP























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Tmax [HJD] error [d] Project Method Notes Tmax [HJD] error [d] Project Method Notes

2454101.7529 0.0005 SuperWASP CCD CCD-144, WASP 2455661.5027 0.0002 Liakos & Niarchos (2011) CCD V

2454111.7874 0.0005 SuperWASP CCD CCD-144, WASP 2456629.5835 0.0003 this paper CCD 24-inch, y


2454115.6921 0.0009 SuperWASP CCD CCD-143, WASP 2456726.6161 0.0005 this paper CCD 1-inch, green








2454143.5770 0.0005 SuperWASP CCD CCD-144, WASP













2454169.7852 0.0007 Pi of the Sky CCD (2006-2009), Clear






















Notes. WASP – SuperWASP observations were unfiltered in 2004, since 2006 broadband filters were used (Butters et al. 2010).

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New Analysis of the Light Time Eﬀect in TU Ursae Majoris · New Analysis of the Light Time...

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