R Scuti: Close Alternating Pulsation Periods or Chaos in the RV Tau -- Type Star?
11 R Scuti: Close Alternating Pulsation
Periods or Chaos in the RV Tau – Type
Star?
L.S. Kudashkina, I.L.Andronov
Department ”Mathematics, Physics and Astronomy”Odessa National Maritime University, Odessa 65029, Ukraine
Abstract
Results of analysis of 60 010 data photometric observations from the AAVSO interna-tional database are presented, which span 120 years of monitoring. The periodogramanalysis shows the best fit period of 70.74d, a half of typically published periods forsmaller intervals. Contrary to expectation for deep/shallow minima, the changes be-tween them are not so regular. There may be series of deep (or shallow) minima withoutalternations. There may be two acting periods of 138.5 days and 70.74, so the beatmodulation may be expected. The dependence of the phases of deep minima arguefor two alternating periods with a characteristic life-time of a mode of 30years. Thesephenomenological results better explain the variability than the model of chaos.
Introduction
R Sct = BD-05 4760= IRAS 18448-0545= HIP 92202. This star belongs to RVTauri - type variables (RVA). RV Tauri stars are generally considered to be post-AGBstars with low initial masses (Jura, 1986). They are metal-poor supergiants of interme-diate spectral type which show a pulsation variability with a light curve characterizedby alternating deep and shallow minima (Mantegazza, 1991). The abundance ratiosshow that they have experienced first dredge-up at the bottom of the red giant branch.Based on their infrared dust excesses, RV Tau stars are classified into two groups: thosewith extensive warm dust and those without evidence of dust in the near-infrared re-gion. R Sct is the brightest star in the visible in the latter group. R Sct has a reportedperiod of 147 days. The effective temperature varies from 4750 to 5250 K; the spectraltype may vary as late as M3 at minimum phase (Matsuura et al., 2002).First paper on the star, which is listed in the ADS, is dated 1890, and is devoted tothe description of the spectrum (Espin, 1890). The GCVS (Samus et al., 2017) providesephemeris with different periods for 15 light curve intervals. The smallest value of theperiod that occurs is 137 . d , and the largest is 152 . d . a r X i v : . [ a s t r o - ph . S R ] F e b .S. Kudashkina, I.L.Andronov 2 Observations
For the analysis, we have used the data published by the American Association ofVariable Stars observers (AAVSO, Kafka 2020). We have used long-term photometricalobservations - visual ones and that with the filter V. After the cleaning the data by re-moving outliers, 60 010 data points remained in the range JD 2414459.4 – 2458470.851(1898–2018), totally, 120 years of observation. As the star shows complicated photo-metric behaviour typical to the RV Tau-type, it got this classification (Samus’ et al.,2017).Earlier, we have already investigated photometric behavior of several RV Tauristars. In total, more than 40 objects of the RVA and RVB types were studied, theirperiodogram analysis was carried out (Kudashkina et al., 1998; Kudashkina, 2019; Ku-dashkina, 2020a), the mean light curves (Kudashkina,2020b; Kudashkina, 2020c) andphase portraits (Kudashkina & Andronov, 2017a; Kudashkina & Andronov, 2017b)were plotted, the behavior of the mean brightness, amplitude and phase with time(Kudashkina et al., 2013). For almost all the stars studied, fairly regular mean cu-rves were obtained over the entire observation interval, which were then smoothed bya trigonometric polynomial, and thus an atlas of mean curves was created (Andro-nov & Chinarova, 2003), similar to the atlas of mean curves for Mira Ceti-type stars(Kudashkina & Andronov, 1996). But not in the case of R Scuti! For this object, themaximum peak on the periodogram does not reflect the star’s variability over the entireobservation interval, and the average light curve with this period is very ”smeared”.
Periodogram Analysis
Periodogram Analysis was made using the trigonomeric polynomial least squareapproximations of orders s = 1 , S ( f ) is the ratio of thevariance of the approximation to the variance of observations (Andronov 1994, 2003,2020; Andronov et al., 2020). The algorithm was realized in the software MCV (An-dronov and Baklanov, 2004). x ( t ) = C + s X j =1 ( C j cos( jωt ) + C j +1 sin( jωt )) . (1)Few values of s are used to test for possible multi-harmonic periodic variations.The results of the periodogram analysis are shown in Fig. 2. The best period forthe entire data set is 138 . d for s = 1 . The peak for s = 4 corresponds to the elements M in.J D = 2447423 .
07 + 414 . · E (2)The corresponding phase light curve is shown in Fig. 3. It shows drastic phase shifts,whereas the approximation corresponds to a stable light curve. As there was a peakclose to 70 d , which is 6 times smaller than the apparent value 414 . d , we also madethe approximation with s = 6 . The corrected period is P = 415 . ± . . d , and themean minimum is shifted by 0 . P. Thus both curves are shown in Fig. 3.However, it is not possible to obtain a good mean light curve with this period. Inaddition to the 70 . d period, which is close to the 2: 1 (1.96) ratio, there is also a peakat the 276 .
68 period value, which is a doubling of the formal period and is 3.91 with R Scuti
Figure 1:
Observations of R Sct from the AAVSO database (blue circles) andits approximations using the trigonometric polynomial of order s = 4 (TP4, up),running parabola (RP, middle), local approximations (LA, bottom). Figure 2:
Periodogram S ( f ) of R Sct obtained using the trigonometric polynomialof orders s = 1 ,
2, 4. The highest peaks are marked with corresponding periods. Thehorizontal red lines show pairs of peaks with an integer ratio 2 or 3. .S. Kudashkina, I.L.Andronov 4
Figure 3:
Phase light curves for the trigonometric polynomial approximations ofdegree s = 4 (left) and s = 6 (right). Observations are shown as circles, the appro-ximations - by lines, Figure 4:
Left: The scalegrams σ , σ [ x C ] , S/N. Right: Λ(∆ t ) scalegram. The num-bers correspond to the filter half-width ∆ t (left) and ”effective periods” (right). a 70 . d period. We have not found anywhere in the literature mentions of a periodtwice as long as the formal one. However, with this longest period, the best mean lightcurve is obtained. It is clearly seen how the phase of deep brightness minima changes(Fig. 3). Scalegram Analysis
Scalegram analysis was made using the algorithm of the weighted running para-bola approximation (RP) introduced by Andronov (1987) and extended by Andronov(2003). The corresponding test-functions are shown in Fig. 4. The maximum of theS/N=SNR (signal-to-noise ratio) occurs at ∆ t = 32 d . The corresponding approxima-tion is shown in Fig. 1. Contrary to the multi-harmonic approximation with a constantperiod and constant shape of the phase curve, the RP approximation follows signifi-cant changes of the amplitude. The analysis of the light curve shows deep and shallowminima, as well as humps.The Λ − scalegram Andronov (2003) shows three peaks corresponding to ”periods”of P Λ1 = 70 d (and a corresponding effective semi-amplitude R Λ1 = 446 mmag), P Λ =135 d ( R Λ = 460 mmag). Much weaker peak corresponds to P Λ3 = 284 d , R Λ3 = 274mmag. R Scuti Times of Minima
Looking for complicated behaviour of the light curve, we have made determinationof individual extrema. For this, we have used the software MAVKA (Andrych 2020,The near-extremal intervals were marked, with a total number 609.These data were approximated with a polynomial of statistically optimal degree,similar to the compilation of the catalogue of the individual characteristics of pulsa-tions of semi-regular stars (Chinarova & Andronov, 2000). Other methods like ”theasymptotic parabola” (Marsakova and Andronov 1996, Andronov, 2005; Andrych etal. 2015) and ”parabolic spline” (Andrych et al. 2020a). (Andrych et al. 2020b) testedeffectivity of different methods. As the observations are distributed very irregularly,many of the extrema have short intervals. So finally we decided to use a single method- polynomials, which are characterized by smaller number of parameters.The dependence of the height of the maxima and the minima on time is shown inFig. 5. It shows relatively smaller scatter for the maxima from 3 . m to 6 . m and alarger one from 5 . m to 8 . m for the minima.These ranges partially overlap, so the data are shown at different panels. Moreover,one may suggest a binary character of the brightness of minima. So we classify theminima as faint ( m › . m ) and bright ones.The time intervals between the maxima δt max (separately, also minima δt min ) areshown in Fig. 6. The range of values is 18 d − d and 16 d − d for the maximaand minima, respectively. The upper limit is due to the gap of observations more thana century ago, during and near the World War I, and thus has no physical meaning.The intervals show condensation to the values P ∼ d and its integer multipliers.However, there are many points between the expected equidistant horizontal lines. The”multiple” periods at this diagram are due to missing extrema - some because they arewithin error corridor of the observations, some because of missing observations due toinvisibility of the star at the day. The scatter is rather large, showing phase drifts ofthe individual cycles.As the observations show such strong period/phase changes, we apply anothermethod.At first, we have decreased the number of minima timings to 129, using only”deep” minima with a brightness m › . m . The ”shallow” minima may be partiallyhidden/biased by observational errors. Additionally, we have used seven moments ofminima published by H¨ubscher (2011). They are close to our results, and unfortunatelyfill no gaps in the AAVSO data. Similarly, we tried to fill the gap with ”shallow” minimanear JD 2437000. They show larger scatter, and do not improve cycle number count.The individual time intervals δt were divided by a preliminary value of the period P = 71 . d . If these (non-integer) cycle counts δE = δt/P were ‹ , the values δt weresummed, as well as the rounded values δJ = int( δE + 0 . . The ratio of these sumshad given another estimate of the period P δ = P ( δt ) / P ( δJ ) = 12401 . /
175 =70 . d , close to the value from the periodogram. The values of δt are shown in Fig. 6.One may see apparent lines, what argues in a systematic period difference fromthat obtained. One may note a ”cell”-type structure of the phases. We tried to changethe trial period with a small step, to see the structure of the phase changes. Taking intoaccount that the phase may cross the limiting values -0.5 and +0.5, we have shown the.S. Kudashkina, I.L.Andronov 6 Figure 5:
The brightness at maxima and minima vs. time. While the maxima mayshow a smoth wave with ∼ , d , the minima show different types (”shallow anddeep”, or ”bright and faint”). The violet line shows a border between these twotypes. Figure 6:
Dependence of the time intervals δt for the maxima and minima. Thehorizontal red lines are multipliers of 70 d , an approximate period. For a constantperiod nad observations without gaps, all the points should be at a horizontal linecorresponding to the basic period. The multiplied values correspond to absent (orvery shallow) extrema. diagram in triple, formally increasing the phase from -0.5 to 1.5. Thus any minimumis shown in triple with a shift of 1.Finally, we adopted a value of P = 70 . d It was used to compute the phases, evenif the usually assumed period is twice larger. The initial epoch was arbitrary set to themoment of the first detected deep minimum:
M in.J D = 2414814 . . · E. (3)In Fig. 7, the phases are shown according to the ephemeris (Eq. (3)). For com-parison, we have shown ”artificaial” moments of minima computed according to thetable of interval-based ephemerids listed in the GCVS. While they are in a reasonablecoincidence with our results, the ephemeris after JD 2445000 is very different for theobserved minima, possible, due to a miscomputation of the number of cycles.The red line shows the period P = 70 . d , which corresponds to the highest peakat the periodogram for s = 1 , which is also seen for other s. The decreasing black linecorresponds to a half of the second significant peak of the periodogram of 138 . d , also R Scuti Figure 7:
Dependence of phases of deep (green and blue points) and selected shallow(black) minima. Red circles in the narrow interval JD 2454604-55016 correspond to7 data points compiled by H¨ubscher (2011) and do not contradict our findings. Forsuitability of looking for phase jumps, these data are shown in triple marked at Fig. 2. These two ”periods” are not very good approximations because ofphase shifts.The strange situation with two ”periods” argues for an absence of the true period.As the next approximation, one may suggest that one period ( ∼ . d ) may be switchedto another ( ∼ . d ) and vice versa. Discussion
As noted by some authors, R Sct is one of the most irregular stars in its classof pulsating variables. R Sct has such large variations in the depth of the minimumsthat it is often difficult to determine in the cycles of the formal period which mini-mums are primary (main) and which are secondary. Gillet (1992) writes that for RSct each frequency peak shows a multi-component structure which is not encounteredwithin power spectra deduced from hydrodynamic stellar models.This means that thepulsation of R Sct is never strictly periodic.The multi-component structure of the frequency signal is characteristic of manystars. However, among this class of objects there are those that show practically ”cle-an” peaks in a frequency ratio of 2: 1. According to the type of periodograms, wepreviously divided the stars into three groups: group I includes objects showing theperiodogram typical form of RV Taurus stars, and the ratio of the periods of the twomain peaks is indicated. Group II includes objects whose periodograms contain signsof multiperiodicity (Multi-p) or vice-versa, only one clear peak, instead of two (Single-p). Group III includes objects whose periodograms are highly noisy mainly due to thesmall number of observations. They do not show the typical details of RV-type stars.(Kudashkina L.S., 2020b; Kudashkina L.S., 2020a). According to this classification, RSct can be placed in at least two different groups (1 and 2), depending on the intervalof its light curve.For the appearance of stochastic behavior with a small number of excited modes,two factors are required: resonant coupling of modes and nonlinear phase drift thatinterferes with mutual synchronization. Also, the linear coupling of modes with closefrequencies plus the inertia of the medium can lead to stochastic behavior (Rabinovich,1978). These conditions can be realized in R Sct, if, as indicated by a number of.S. Kudashkina, I.L.Andronov 8authors, this star is still on the AGB in the stage of a thermal-pulsation cycle of flarecombustion of helium (Matsuura, M. et al., 2002). An additional inflow of energy froma flare violates the stationary self-oscillation regime, bringing the star into a state oflow-dimensional deterministic chaos, that is, into a state on the verge of disappearanceof regular self-oscillation regimes.On the other hand, the violation of the regularity of the oscillations can be causedby the complex interaction of shock waves propagating in the photosphere and theextended atmosphere of the star. These effects can also be superimposed on the impactfrom the dust envelope (Gillet et al., 1989; Yudin et al., 2003). We previously studiedthe influence of shock waves on the light curve for long-period variables that are locatedon the AGB and, possibly, are closely related to RV Tauri - type stars (Kudashkina &Rudnitskij, 1994, 1988). It is also impossible not to take into account random changesin the period, for example, as a result of a change in the convection mode.
Conclusion
Our work is based on R Sct observations published by the American Associationof Variable Stars observers (AAVSO, Kafka 2020) for 120 years.The periodogram analysis was carried out. The best period for the entire data setis 138 . d for s = 1. For s = 4 elements for a phase light curve with a period 414 . d are obtained. Also, this period was refined to the value 415 . d .The scalegram analysis was carried out. Three peaks were revealed correspondingto the values of the periods 70, 135 and 284 days.Thus, the main period is interrupted with phase shifts. Due to this, arises theapparent conclusion that the star has two periods and switches between them.An alternate model for variability of R Sct is due to non-linear chaos (Buchlerand Koll´ath, 2003). It shows beat-like nearly periodic modulation of amplitudes andphases. Our analysis argues for ”switches” between close periods with a characteristictime of ∼ Acknowledgements.
We acknowledge with thanks the variable star observationsfrom the AAVSO International Database contributed by observers worldwide and usedin this research. This work is in a frame of the international projects ”Inter-longitudeastronomy” (Andronov et al. 2003) and ”Astroinformatics” (Vavilova et al, 2016).