Period changes in the RR Lyrae stars of NGC 6171 (M107)
A. Arellano Ferro, P. Rosenzweig, A. Luna, D. Deras, S. Muneer, S. Giridhar, R. Michel
aa r X i v : . [ a s t r o - ph . S R ] J a n Astronomische Nachrichten, 16 January 2018
Period changes in the RR Lyrae stars of NGC 6171 (M107)
A. Arellano Ferro , P. Rosenzweig , A. Luna , D. Deras , S. Muneer , Sunetra Giridhar ,R. Michel Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico. Ciudad Universitaria, CP 04510,M´exico: ([email protected]) Grupo de Astrof´ısica Te´orica, Facultad de Ciencias, Universidad de Los Andes, Venezuela. Indian Institute of Astrophysics, Koramangala, 560034, Bangalore, India Observatorio Astron´omico Nacional, Instituto de Astronom´ıa Universidad Nacional Aut´onoma de M´exico,Ap. P. 877, Ensenada, BC 22860, M´exicoReceived 2017, accepted 12-01-2018Published online XXXX
Key words globular clusters: individual (NGC 6171) – stars:variables: RR LyraeBased on photometric data obtained between 1935 and 2017, O − C diagrams were built for 22 RR Lyrae starsin the globular cluster NGC 6171, leading to the discovery of secular period changes in 4 variables for whichwe have calculated their period change rates β . In contrast we find that 82% of the sample stars have stableperiods over the last 82 years. For the stable period stars, the whole data base has been employed to refinetheir periods. Among the period changing stars, three (V10, V12 and V16) have decreasing periods largerthan expected from stellar evolution. Despite these individual cases of significant period change rate, thegolbal average of the measured period changes in the cluster is basically zero, in consonance with theoreticalpredictions for clusters with redder horizontal branches. The hitherto unpublished observations, now broughtinto public domain, are employed to calculate a set of times of maximum light which are used in the presentanalysis. Copyright line will be provided by the publisher
The study of secular period changes of RR Lyrae stars(RRLs) in globular clusters, may play a decisive role intesting horizontal branch (HB) evolution models. How-ever, measuring secular period changes from observa-tions is difficult since accurate observations over a verylong time-base are required. Only a few clusters havebeen studied from data covering more than 60 years,e.g.; M3 (Corwin & Carney 2001, Jurcsik et al. 2012),M5 (Szeidl et al. 2011; Arellano Ferro et al. 2016),NGC 6934 (Stagg & Wehlau 1980), M14 (Wehlau &Froelich 1994), M15 (Silbermann & Smith 1995), NGC7006 (Wehlau, Slawson & Nemec 1999), and ω Cen(Jurcsik et al. 2001). Theory predicts that blueward orredward evolution near the zero-age horizontal branch(ZAHB) is slow and produces very small period changerates, except towards the end of the HB evolution, whenthe values of β = ˙ P can be between +0 . .
15d Myr − (Lee 1991). However, in several of the abovestudies, stars with significantly large values of β , bothpositive and negative, have been reported. These highvalues may be the result of non-evolutionary effects,like stochastic processes related to mixing events in thecore of the star (Balazs-Detre & Detre 1965, Sweigart &Renzini 1979), or to the fast crossing of the instability strip of pre-ZAHB stars on their evolution to the blue,with β ≤ − . − (Silva-Aguirre et al. 2008).In the present investigation we have focused ourattention on the globular cluster NGC 6171 (M107 orC1629-129 in the IAU nomenclature, α = 16 h m . s , δ = +13 ◦ ′ . ′′ J2000, galactic coordinates l = 3 ◦ . b = +23 ◦ . ∼
82 years, constitutes a significant im-provement and encourages a new approach to the studyof the secular period changes of the RRLs of this clus-ter.The photometric data were used to calculate asmany times of maximum light as possible and thesewere employed to investigate the secular period behav-
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Arellano Ferro et al.: RR Lyrae period changes in NGC 6171
Table 1: Sources of photometric data of the RRLs inNGC 6171.
Authors years bandOosterhoff (1938) 1935 pg Coutts & Sawyer Hogg (1971) 1946-1970 pg Kukarkin (1961) 1959-1960 pg Mannino (1961) 1959-1960 pg Dickens (1970) 1966-1967 B pg and V pg Table 3 (this paper) 1972-1991 B pg Clement & Shelton (1997) 1993-1994 CCD V Table 2 (this paper) 2015-2017 CCD V Table 2: 2015-2017
V I photometry data of the RRLsin NGC 6171. A full version of this table is available inelectronic format (see Supporting Information).
Variable Filter HJD M std σ m Star ID (d) (mag) (mag)V4 V 2457200.14416 15.673 0.006V4 V 2457200.14769 15.667 0.006... ... ... ... ...V4 I 2457200.13190 14.797 0.009V4 I 2457200.13824 14.787 0.009... ... ... ... ...V5 V 2457200.14416 15.287 0.004V5 V 2457200.14769 15.286 0.004... ... ... ... ...V5 I 2457200.13190 14.327 0.007V5 I 2457200.13824 14.302 0.007... ... ... ... ... ior of 22 RRLs in NGC 6171. The sources and temporaldistribution of the data are indicated in Table 1.The present paper is organized as follows; in § § § O − C method is described. In § O − C diagrams and the resulting refined periods andperiod change rates. In § § The most recent V CCD time-series observations usedin this paper substantially extend the time baseline,in many cases to ∼
82 years. These observations wereperformed on 4 nights, between June 26, 2015 to May5, 2016, with the 2.0 m telescope at the Indian Astro-nomical Observatory (IAO), Hanle, India. For 7 nightsbetween June 29 to July 5, 2017, data were obtainedwith the 0.84 m telescope at the San Pedro M´artir Ob- servatory (SPM), M´exico. A total of 292 images ob-tained in the Johnson-Kron-Cousins V filter, are usedfor the purpose of the present analysis. I images werealso obtained. The instrumental system was convertedinto the Landolt-Johnson/Kron-Cousins standard sys-tem via standard stars in the field of the cluster pro-vided by Stetson (2000). The standard system V I mag-nitudes and their uncertainties for the RRLs are pub-lished in electronic format and we present a small por-tion of it as Table 2.
The data taken between 1935 and 1994 have been sys-tematically assembled by Prof. Christine M. Clement,who kindly made them available to us. The originalsources are summarised in Table 1. The published datahave been taken as given in the original papers withoutany further manipulation. The observations from theyears 1972-1991 were obtained by C. Clement with theUniversity of Toronto 61-cm telescope at the Las Cam-panas Observatory of the Carnegie Institution of Wash-ington. A total of 420 photographs were taken on plateswith 103aO emulsion, exposed through a GG385 filter.The plates were measured on a Cuffey iris photome-ter. Some of these data were used, but not published,in the study of the Fourier parameter φ , by Clementet al. (1992). We are now publishing these data in anelectronic format and a small fraction of the table isincluded in the printed version of this paper as Table3. These data were taken in an almost yearly basis andthe light curves of most variables are well covered, al-lowing a good estimation of the time of maximum lightnearly every year. Thus, this set of data is crucial forthe interpretation of the O − C diagrams. In Fig. 1 the V light curves tabulated in Table 2 andobtained in 2015-2016 (black symbols) and 2017 (bluesymbols) are displayed. They have provided a few re-cent times of maximum light that extend the time-baseto 82 years. They have been phased with corrected pe-riod estimated below and the seasonal time of maxi-mum. Note the consistency of the 2015-2016 and the2017 data and the evident variation in amplitude insome stars, likely due to the Blazhko effect. The em-ployment of these light curves as indicators of the meanmetallicity and distance of the parental cluster, viathe Fourier decomposition, will be reported elsewhere.Comments on individual stars can be found in § O − C approach to the secularperiod changes The observed minus the calculated ( O − C ) residualsof a given feature in the light curve, as an indication of Copyright line will be provided by the publisher sna header will be provided by the publisher 3
Table 3: Las Campanas photographic data (B mag) taken between 1972 and 1991 by C. Clement and previouslyunpublished. This is an extract from the full table, which is only available with the electronic version of thearticle (see Supporting Information).
HJD V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12-2 400 00041446.819 16.86 16.44 15.56 15.91 16.02 15.33 16.44 15.77 16.84 16.07 16.8941447.640 – – – – – – – – – – –41447.676 15.84 15.89 15.56 16.29 15.60 16.29 15.25 – 16.86 16.61 16.61. . .HJD V13 V14 V15 V16 V17 V18 V19 V20 V21 V23 V24-2 400 00041446.819 16.79 16.89 15.66 16.84 16.66 15.24 15.91 16.70 16.95 16.15 –41447.640 – – 15.77 16.05 15.77 15.99 16.44 – – 15.89 –41447.676 16.98 16.73 15.79 16.42 15.93 16.02 15.89 16.02 17.16 15.73 16.29. . . Fig. 1: Light curves of the 2015-2017 data, except V23 for which the data from Clement & Shelton (1997) wereplotted since this star is not in the field of the 2015-2017 data. Black symbols are used for the 2015-2016 datafrom Hanle. Blue symbols are for the 2017 data from SPM. The light curves were phased with the new periodslisted in Table 5 (except for V5) and the local epochs of the maximum light for a given data set as listed in Table4, see § Copyright line will be provided by the publisher
Arellano Ferro et al.: RR Lyrae period changes in NGC 6171 miscalculations or authentic variations of the pulsationor orbital period, using a single given phase of the lightcurve as a reference, is a standard approach that hasbeen in use for many decades; for example, in Cepheids,RR Lyrae stars, and contact binary stars (e.g. ArellanoFerro et al. 1997; 2016; Coutts & Sawyer Hogg 1969).Then, it is convenient to select a feature that facilitatesthe accurate determination of the phase. For RRLs, themaximum brightness is a good choice since it is wellconstrained, particularly for the RRab type, as opposedto the longer-lasting time of minimum. To predict thetime of maximum let us adopt an ephemeris of the form C = E + P N, (1)where E is an adopted origin or epoch of reference, P is the period at E , and N is the number of cycleselapsed between E and C . An initial estimate of thenumber of cycles, between the observed time of maxi-mum O and the reference E is simply N = (cid:22) O − E P (cid:23) , (2)where the incomplete brackets indicate the roundingdown to the nearest integer. However, it must be notedthat if the time between E and the observed time ofmaximum O is much larger than the period, and theperiod change rate is large enough, the O − C differencecan exceed one or more cycles, and there must be acorrection for these extra cycles to obtain a correct O − C diagram. This exercise may prove to be difficult ifthere are large gaps in the time-series, but it is ratherstraight forward otherwise.A plot of the number of cycles N vs O − C , is usu-ally referred as the O − C diagram, and its appearancecan make evident secular variations of the period, orthe fact that the period P used in the ephemerides iswrong, in which case the distribution of the O − C resid-uals is linear and tilted. For NGC 6171, the distributionof the observations over the last 80 years enabled us toestimate accurately the number of cycles required inorder to produce coherent O − C diagrams.Let us assume, as an initial model, a quadratic dis-tribution of the O − C residuals as a function of time,represented by the number of cycles N elapsed sincethe initial epoch E . The linear and quadratic casesare then particular solutions of the representation: O − C = A + A N + A N , (3)or, O = ( E + A ) + ( P + A ) N + A N . (4)Taking the derivative, the period at any given N is P ( N ) = dOdN = ( P + A ) + 2 A N. (5) Table 4: Observed times of maximum light, O , for theRRLs in NGC 6171 and their corresponding O − C residuals calculated with the given ephemerides foreach variable. The sources of either the times of maxi-mum light or the data employed to calculate them, arecoded in column 4 as follows: Oo (Oosterhoff 1938),CouSH (Coutts & Sawyer Hogg 1971), Man (Mannino1961), Kuk (Kukarkin 1961), Di (Dickens 1970), Clem(Las Campanas, this paper), ClSh (Clement & Shel-ton 1997), Tab3 (Hanle and SPM, this paper). This isan extract from the full table which is available in theelectronic version of the article (see Supporting Infor-mation). Variable P ( days ) E (HJD)V2 0.571021 2442538.8397 O (HJD) O − C No. of Cycles source(days)2427930.951 –0.0346 –25582. Oo2432004.682 0.0340 –18448. CouSH2437052.486 0.0142 –9608. Kuk2437100.448 0.0104 –9524. Man2439258.334 0.0088 –5745. Di2440693.874 0.0025 –3231. CouSH2441448.747 –0.0140 –1909. Clem2442135.701 0.0020 –706. Clem2442537.670 –0.0276 –2. Clem2442538.841 0.0013 0. Clem2442935.727 0.0279 695. Clem2443273.736 –0.0075 1287. Clem2443632.899 –0.0165 1916. Clem2444020.635 –0.0037 2595. Clem2444371.805 –0.0115 3210. Clem2444759.535 –0.0046 3889. Clem2445083.888 0.0086 4457. Clem2445850.780 0.0197 5800. Clem2446562.828 0.0047 7047. Clem2447298.862 -0.0071 8336. Clem
From the above equations, it is straight forward todemonstrate that the period change rate β = ˙ P at N = 0 and P = P is given by β = β = 2 A P , (6)and that if the O − C distribution is linear, i.e. A = 0,then the correct epoch and period are given by E + A and P + A , respectively. A detailed derivation of theabove equations can be found in Arellano Ferro et al.(2016). O − C diagrams We have estimated as many times of maximum bright-ness as possible with the available data. When a lightcurve is covered near the maximum, estimating thetime of the maximum brightness is fairly straight for-ward and the uncertainty is small. Error bars would besimilar in size to the symbols in the O − C diagrams.Only in a few cases, interpolating between competing Copyright line will be provided by the publisher sna header will be provided by the publisher 5
Table 5: New periods and period change rates for RRLs in NGC 6171. The uncertainties in β correspond to theuncertainty in the coefficient A in eq. 6. Variable Variable P E E + A P + A β Star ID Type (+2 400 000) (+2 400 000) ( O − C )(days) (HJD) (HJD) (days) (d Myr − )V2 RRab 0.5710 42538.841 42538.840 0.571021 − . ± . . ± . . ± . − . ± . . ± . − . ± . . ± . . ± . − . ± . V11 RRab 0.5928 57528.3744 57528.3821 0.592809 +0 . ± . − . ± . V13 RRab 0.4668 44371.865 44371.873 0.466797 +0 . ± . . ± . . ± . − . ± . V17 RRab 0.561168 41860.578 41860.593 0.5611675 + . ± . V18 RRab 0.561404 57528.4361 57528.4361 0.561404 +0 . ± . . ± . − . ± . − . ± . . ± . − . ± . maxima was necessary. For the data sets covering sev-eral years (e.g. Coutts & Sawyer Hogg 1971 for 1946-1970 or data in Table 3 for the years 1972-1991), wesearched for clear times of maximum through the wholecollections and were able to recover numerous maxima,producing the highest density of data in the O − C diagrams in Fig. 2.The complete collection of times of maximum lightis given in Table 4. To calculate the O − C residuals weproceeded as follows: first, an epoch with a well coveredlight curve was identified and the period at that epochwas estimated. These period and epoch were adoptedas initial values P and E . Generally, the data from2015 to 2016 were proper for this aim, except for afew incomplete light curves particularly near the timeof maximum. In those cases, the data from Clement& Shelton (1997) or from the previously unpublisheddata from Clement (Table 3), were used. For some caseswith a linear distribution of O − C values, we took theperiod from the Catalogue of Variable Stars in GlobularClusters (CVSGC) (Clement et al. 2001; 2015 edition).While these periods are quoted to only four digits andtrue periods may be slightly different, the approach tothe period correction is not sensitive to the selection of P since a different P will produce a different slope A but the corrected period P + A (see Eq. 5) will be thesame. The adopted initial ephemerides are summarisedin Table 4. The resulting O − C diagrams are shown inFig. 2 for every variable included in the present work. It is obvious from these diagrams, that there areonly two types of O − C distributions; the linear dis-tribution, which once the correct period is used in thetime of maximum predicting ephemerides (given in col-umn 5 of Table 5), produces a horizontal distributionand implies a non-changing period, and the parabolicdistribution which implies a secular period variationwhose rate can be calculated with Eq. 6. It is worthnoting at this point, that the O − C approach to thesecular period changes is particularly sensitive to thecounting of cycles, which is very easy to lose when deal-ing with a long time-base and short-period stars, as isthe present case. As the O − C difference drifts, eitherbecause the assumed initial period is wrong or becauseit is authentically changing, the calculated maximummay skip a few cycles relative to the corresponding ob-served one. If this is not properly considered, the O − C diagram may show intriguing shapes which could bemisinterpreted as irregular period variations. This hasalready been stressed by Arellano Ferro et al. (2016)for the case of the RRLs in M5. In the present case ofNGC 6171, we do not see any irregular variations butonly constant periods (linear) or a few parabolic sec-ular variations (quadratic). It is pointed out however,that there are cases where the horizontal O − C dis-tributions display a considerable scatter (e.g. V6, V7,V18, V21). This may be the consequence of stochasticfluctuations of the period and/or of uncertainties in the Copyright line will be provided by the publisher
Arellano Ferro et al.: RR Lyrae period changes in NGC 6171 Fig. 2: O − C residuals as a function of number of cycles (filled circles). For stars with linear distributions, the O − C residuals were calculated using the refined ephemerides (columns 5 and 6, in Table 5). The O − C residualsplotted with a triangle, filled and empty when two solutions were tried, stand out from the rest of the distributionand were not considered in the adopted solutions. Solid lines represent the finally adopted period change solutionfor cases with β different from zero. Dashed horizontal lines are plotted at ( O − C )=0 as a reference. In thecase of V5 we explicitly show the linear distribution showing the constancy of its period. An apparent abruptperiod change in V14 is illustrated by the empty circles and two linear solutions. These and a few cases withpeculiarities are discussed in more detail in § Copyright line will be provided by the publisher sna header will be provided by the publisher 7 estimations of the times of maximum light due to thelimited quality of the observations.Table 5 summarises the initial assumed ephemerides,the corrected periods and epochs and, in the corre-sponding cases, the calculated period change rate β . Once a period change rate has been calculated (quadraticcase), or the period has been duly corrected (linearcase), a natural test is to phase the light curve withthe new ephemeris. In Fig. 1, the light curves from2015-2016 are phased either with corrected period (lin-ear cases) or with P ( N ) for the corresponding N = 0in Eq. 5 (parabolic cases).In fact, for a secularly changing period, Eq. 5 allowsto calculate the period at any given number of cycles N elapsed from the origin E . Each corresponding periodshould properly phase the data taken at that vlue of N . As an example, in Fig. 3 we phase the light curveof star V12 over the last 82 years, using the ”local”periods and epochs as predicted by the parabola in Fig.2 and Eq. 5, and listed in Table 6. In all cases, the lightcurve and time of maximum are consistent with thesecular period change. The appearance of the curve isonly limited by the quality of the photometry of a givendata set. While generally the O − C diagrams in Fig.2 show aclear period behavior, in a few cases the O − C distribu-tion may admit alternative solutions, as we comment inthe following paragraphs. Since the previous study ofthe period changes in NGC 6171 by Coutts & SawyerHogg (1971), good quality data have been obtained,and hence richer O − C diagrams can be produced, adetailed comparison with the results of these authors isprobably inadequate. Some comments on specific starsmight however be in order.V2 and V3. These RRLs are not included in the fieldof view of our images and the historical data are scarce.However, the data in Table 3 enable the estimationof numerous times of maximum light and hence theanalysis of the secular behaviour of the period.V5. The O − C residuals in Fig. 2 display a clear lin-ear distribution, leading to a refined period of 0.702376d. A negative period change rate and an abrupt nega-tive period change have been reported for this star byCoutts & Sawyer Hogg (1971) and Gryzunova (1979a)respectively. Our solution displays a rather constant pe-riod. The estimated maximum from 2015 data shows asignificant and unexpected phase displacement and wasnot considered in the adopted solution. We noted how-ever that the predicted period in Table 5 (0.702376 d)fails to phase properly the light curves from 2015-2017, for which a shorter period, 0.695248 d, had to be in-voked to scale the curves well. We do not have a clearexplanation for this behaviour and speculate that thestar might have undergone a stochastic variation of itsperiod. The amplitude variation between 2015-16 and2017 should be noted.V6. Although the O − C distributions of this starsuggest a linear distribution, the scatter is significantand is probably due to the bump near maximum whichmakes the estimation of the time of maximum bright-ness inaccurate.V7. In our data the light curve of this star displaysa large difference between 2015-2016 and 2017 and sug-gests a large amplitude modulation as observed in starswith the Blazhko effect. Clement & Shelton (1997) no-ticed the cycle to cycle variations near minimum light,and the peculiar harmonics amplitude rations relativeto other RRab stars in the cluster. Stars with Blazhkoeffect often display not only amplitude but also phasemodulations, which likely explains the large scatter inthe O − C diagram for this star.V10. A period increase was reported by Coutts &Sawyer Hogg (1971) who calculated β = 1 . − .The diagram in Fig. 2 shows a decreasing nature of theperiod at a rate β = − . ± .
039 d Myr − . Thediscrepancy between the two investigations is causedby an error in the period due to an uncertainty in thenumber of cycle counts in the CSH study. The richer O − C diagram in our current study has resolved thisambiguity.V12. Our analysis of this star shows a decreasing pe-riod with β = − . ± .
168 d Myr − . The O − C dia-gram however displays a significant dispersion. We notethat Clement & Shelton (1997) found the Fourier pa-rameters, particularly φ and φ , to be peculiar amongthose in other RRab stars. A close inspection of Fig. 3shows a distinctive slope change on the rising branchin 1993 which is not apparent in the other light curves.This suggests that the light curve shape might un-dergo secular variations and probably stochastic oscil-lations of the time of maximum. It should be notedthe amplitude variation between 2015-2016 and 2017data, confirming the amplitude modulations reportedby Clement & Shelton (1997).V14. The O − C residuals for this star show a pe-culiar change in slope at about HJD 2443275.6 d orMay 1977 if a period of 0.4816 d is used. Although thisslight change of slope could also be fitted by a parabola,implying β = +0 . ± .
07 d Myr − , we rather preferthe slight period change and hence the two slopes de-picted. The two slopes would imply corrected periods P . P . O − C diagram is found withtwo discrepant values in 2015 and 2017, which are oth- Copyright line will be provided by the publisher
Arellano Ferro et al.: RR Lyrae period changes in NGC 6171 Fig. 3: A selection of light curves of V12 over the past 82 years phased with the corresponding period and epochas predicted by the parabola in Fig 2 and Eq. 5. Note the variations in amplitude. These ephemerides are listedin Table 6.Table 6: A selection of epochs and periods of star V12over 80 years. The phased light curves are shown inFig. 3.
E0 P (days) year2427931.957 0.47289860 19352436728.475 0.47287822 19592437050.549 0.47287747 19602440747.669 0.47286889 19702441454.580 0.4728665 19722443281.546 0.4728627 19772446185.774 0.4728567 19852448011.631 0.4728529 19902449125.719 0.47284945 19932449482.737 0.47284862 19942457527.331 0.47282997 2015-20162457939.681 0.47283244 2017 erwise consistent among them. This may suggest anabrupt period change which is to be confirmed in thefuture if new times of maximum light become available.V16. This is a clear and strongly period decreas-ing star with β = − . ± . − . V17. The O − C diagram in Fig. 2 shows a positiveparabola and a corresponding β = +0 . ± . O − C diagram is substantial, its periodhas been refined and reported in Table 5.V22. This star is not included in the present study.V22 is not a cluster member (Sawyer Hogg 1973) andout of the field of most studies, except of that of Oost-erhoff (1938). Hence the historical data are very scarce.V23. The O − C diagram shows a linear distributionwith a small tilt which implies a tiny correction to theperiod. The 1935 light curve from Oosterhoff (1938) isvery scattered and we could not estimate a reliable timeof maximum. We note that the two oldest maxima, cor-responding to data from Coutts & Sawyer Hogg (1971)from July 1946 and July 1955, are not fitted by themore recent linear distribution, however these are two bona-fide maxima from a rather scattered light curve. Copyright line will be provided by the publisher sna header will be provided by the publisher 9
We note that in the paper by Dickens (1970) the starlabeled as V1 in fact corresponds to V23.
As stars evolve across the instability strip (IS), theirpulsation period should either increase or decrease ifevolution is towards the red or the blue, respectively.However, other non-evolutive reasons for period changeshave been suggested, such as stochastic variations (Balazs-Detre & Detre 1965) or mixing events in the core of astar at the HB that may alter the hydrostatic struc-ture and pulsation period (Sweigart & Renzini 1979).Also, irregular and complicated secular period varia-tions have been claimed for some RR Lyrae stars (e.g.Szeidl et al. 2011, Jurcsik et al. 2001), which indeedwould be difficult to reconcile with stellar evolutionexclusively. However, for the RR Lyrae stars in M5,it has been argued that there is no need to claim for ir-regular period variations since an improper counting ofcycles, particularly in long time baseline sets of timesof maximum light, may be responsible for apparent ir-regularities (Arellano Ferro et al. 2016).At present, there is a large evidence that there isno preferential positive or negative values of β in theRRLs in a given cluster, and for a summary the readeris referred to the discussion of Arellano Ferro et al.(2016). Perhaps, the exceptions are ω Cen for whichan average of β = +0 . ± .
561 d Myr − can becalculated from table 6 of Jurcsik et al. (2001), andIC 4499 with β = +0 . ± .
60 d Myr − from Kun-der et al (2011) (their Table 1 without three extremecases). Also, in the extensive investigations on secularperiod variations in globular clusters (e.g. Silbermann& Smith 1995; Corwin & Carney 2001; Arellano Ferroet al. 2016) no significant differences have been foundfor the average values of β for the populations of RRaband RRc stars .Models of the HB calculated by Lee (1991) andCatelan et al. (2004) confirm that positive period changerates of evolutionary origin occur mostly in globularclusters with blue HB structures, i.e. with large valuesof the HB structure parameter L = ( B − R ) / ( B + V + R ), where B, V and R are the number of stars to theblue, inside and to the red of the instability strip re-spectively. Figure 15 of Catelan (2009) displays suchbehaviour of β as a function of L and shows that inred HB clusters the average value of β should be aboutzero. NGC 6171 has a very red HB, with L = − . β ∼ − . In fact, the overall average of β values in Table5 is − . ± .
346 d Myr − , which given the typi-cal uncertainties of β is not significantly different fromzero. Even in clusters like M3 and M5 where the over-all period change rates average nearly zero, as pre-dicted by canonical models, it has been common toisolate individual cases with significantly large valuesof β . In the case of NGC 6171, we found large secu-lar period change rates in 4 stars in a sample of 22.Three of these have large negative values of β , whichimplies evolution to the blue. The only period increas-ing case is V17 with β = +0 . ± . L =+0.31) for example, V8(+0.474 d Myr − ), V7 (+0.474 d Myr − ) and V25(+0.933 d Myr − ), or the more moderate V77 (+0.340d Myr − ), V87 (+0.369 d Myr − ) and V90 (+0.114d Myr − ) for which, arguments in favour of they be-ing stars in a truly advanced evolution have been of-fered (Arellano Ferro et al. 2016); in M3 ( L =+0.18) wehave V10 (0.385 d Myr − ), V47 (+0.393d Myr − ), V69(+0.414 d Myr − ), V83 (+0.345 d Myr − ) (Corwin &Carney 2001). Jurcsik et al. (2001) calculated periodchange rates across the IS between − .
026 and +0.745d Myr − based on post-HB evolutionary tracks of Dor-man (1992) for [Fe/H] = − .
48 and masses of 0.60 to0.66 M ⊙ . Thus, all the above quoted positive periodchanges may be consistent with an evolutionary origin,also for our present case of V17.On the other hand, as discussed by Silva-Aguirre etal. (2008), pre-ZAHB stars crossing the instability stripat high evolving rates may have values of β ∼ − . − but can reach values < − . − . From thecalculations of Jurcsik et al. (2001) based on Dorman’s(1992) post-ZAHB models, the fastest blueward evolu-tion reaches the rate of − . × − d d − or about − .
026 d Myr − . Thus, it is tempting to suggest thatvariables V10, V12 and V16 in NGC 6171 are examplesof pre-ZAHB stars. We should consider however that,according to the statistics produced by the simulationsof Silva-Aguirre et al. (2008) for the case of M3, only1 pre-ZAHB is expected every 60 bona-fide HB stars,and only 22% of them would fall in the instability strip,where they can be disguised as RRLs. Assuming thatthese statistics hold for NGC 6171, with about 110 starsin the HB, it implies that not even one pre-ZAHB RRLyrae-like pulsator should be found. Nevertheless, RRLyrae with large negative values of β is a rather com-mon feature in several clusters. In M3 itself there are 5stars with β < − . − (Corwin & Carney 2001),in M5 there are 5 stars with β < − . − (Are-llano Ferro et al. 2016) and 4 in NGC 6934 (Stagg &Wehlau 1980). Thus, it is probably not unlikely thatV10, V12 and V16 in NGC 6171 are indeed pre-ZAHBstars.Perhaps the most remarkable result in the presentpaper is the high percentage of stars with stable period.It should be noticed that the four stars with changingperiod in NGC 6171 are RRab stars, and that 18 of the Copyright line will be provided by the publisher
22 stars studied, i.e. 82%, have retained a constant pe-riod for at least 80 years. In M5, 34% of its RRLs werefound to have unchanging periods over a 100 year inter-val (Arellano Ferro et al. 2016). Naturally, the questionof whether this result would be subject to change afterone or two decades of accurate estimation of times ofmaximum may be posed. Note that the uncertaintiesin the beta values in Table 5 are generally of a few hun-dreds of d Myr − , and that, if the linear distributions inFig. 2 are forcibly fitted with a parabola, the quadraticterm leads to very small and non-significant values of β . We conclude then, that with the data on hand, weare unable to detect period variations, positive or nega-tive, below these limits. Thus, period changes for starsevolving very near the HB at very low rates may passundetected. Pulsation period changes have been analysed via thetimes of maximum light for 22 RRLs in NGC 6171.Archival data collected from the literature, previouslyunpublished data spanning 19 years, and recent CCDobservations enable a span of up to 82 years for mostof the sample stars, which makes this work the firstsignificant study of period changes in NGC 6171. Secu-lar period variations were found for 4 stars, three withsignificant decreasing periods and one (V17) with in-creasing period. No signs of irregular period variationswere found in the RRLs of this cluster but instead theyall have either a remarkably stable period or a secularperiod change that can be represented by a parabolic O − C diagram.The overall average of the period change rates foundin NGC 6171 is not significantly different from zero, asexpected from the canonical evolutionary models of theHB for a cluster with a red HB. Not withstanding thisfact, individual stars with large positive and negativeperiod changes have been found, a trend also observedin M3 and M5.In NGC 6171, we have found a single case withpositive β (V17) which seems to be consistent withthe period change rate expected in a truly advancedstage of evolution towards the AGB. On the contrary,a few cases emerged with values of β significantly neg-ative, which cannot be reconciled with post-HB evo-lutionary predictions, and that may be examples ofpre-core-helium-burning stars on their contraction to-wards the ZAHB. The majority of the RRLs in thiscluster, both RRab and RRc, display a stable periodfor at least 82 years well within the uncertainties ofthe O − C approach. Under the paradigm that periodchanges are a consequence of stellar evolution, it mustbe concluded that these stable stars are evolving veryslowly and their putative period changes are, given the data presently available, undetectable by the approachdescribed in this work. Acknowledgments
We are grateful to Prof. Christine M. Clement for en-couraging this work and for supplying us with her col-lection of relevant historical data of NGC 6171, andher own extensive unpublished observations taken be-tween 1972 and 1991 and allowing us to publish themin the present paper. Her comments and suggestionsto the manuscript are gratefully appreciated. We arealso indebted with the anonymous referee for his/heruseful suggestions and enlightening comments. We ac-knowledge the financial support from DGAPA-UNAM,M´exico via grants IN106615-17, IN105115 and fromCONACyT (M´exico). PR is grateful for the financialsupport from the PREI program of the National Uni-versity of M´exico and the hospitality of the Institutode Astronom´ıa (UNAM). PR warmly acknowledges thefinancial support of the CDCHTA - Universidad de LosAndes (ULA) through project C-1992-17-05-B. We arethankful to Carlos Chavarr´ıa for his help with some ofthe obervations in SPM. We have made an extensiveuse of the SIMBAD and ADS services, for which weare thankful.
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