Period Variation Rates of Four Radial Single-Mode High-Amplitude Delta Scuti Stars

Abstract

In this work, we present a study on the long time-scale period variations of four single-mode high-amplitude delta Scuti stars (HADS) via the classical $O-C$ analysis. The target HADS are (i) XX Cygni, (ii) YZ Bootis, (iii) GP Andromedae, and (iv) ZZ Microscopii. The newly determined times of maximum light came from the Transiting Exoplanet Survey Satellite (TESS), American Association of Variable Star Observers (AAVSO), and Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne (BAV) projects. Together with the times of maximum light obtained in the historical literature, the $O-C$ analysis was performed on these HADS, in which we obtained the linear period variation rates $\dot{P}/P$ as $(9.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$, $(3.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$, $(4.22\pm 0.03)\times {10}^{-8}{\mathrm{yr}}^{-1}$, and $(-2.06\pm 0.02)\times {10}^{-8}{\mathrm{yr}}^{-1}$, respectively. Based on these results and some earlier research, we also discuss the evolutionary stages and the mechanisms of the period variation of these four HADS.

1. Introduction

Delta Scuti stars are pulsators located in the classical Cepheid instability strip crossing the main sequence and sometimes in the region between the main sequence and the giant branch. The mass of delta Scuti stars ranges from 1.4 $M_{\odot }$ to 2.5 $M_{\odot }$, and their pulsation period ranges from $0.02$ to $0.3$ days [1]. As a subclass of delta Scuti stars, the High Amplitude Delta Scuti stars (HADS) generally have larger amplitude (≥0.1 mag) and slow rotation speed ($vsini\le 30\mathrm{km}{\mathrm{s}}^{-1}$) [2], which can be Population I or II stars. Most of the HADS exhibit one or two radial pulsation modes, while some of them display three radial pulsation modes and even some non-radial modes [3,4,5,6,7,8]. For those HADS which exhibit only one dominant radial pulsation mode (see, e.g., [9]), we can use the deviation of their times of maximum light (TML) between the observed and calculated ones to study their period variations ($O-C$ analysis).

The $O-C$ analysis provides us an opportunity to study the period variation details of these stars. Theoretically, for HADS, the period variation rates caused by stellar evolution should be in the range of ${10}^{-10}{\mathrm{yr}}^{-1}$ to ${10}^{-7}{\mathrm{yr}}^{-1}$ [10]. If the observed value is of the same order of magnitude as the theoretical one, it indicates that the period variation of this star could be attributed to stellar evolution; if not, the period variation may be influenced by other mechanisms such as nonlinear mode interaction, mass transfer or the light-travel time effect in binary systems [5,6,7,10,11,12,13,14]. Consequently, the investigation into the period variation of HADS will help us have a better understanding of their evolutionary stage and reveal the unknown mechanisms behind it.

In this work, we chose four single-mode HADS as the targets to study their period variation rates, which have been observed by different projects many times in history and recent years. The basic information about these four HADS is listed as follows.

XX Cygni (XX Cyg, $\langle V\rangle ={11}^{\mathrm{m}}.7$, $P_0\approx 0.1349\mathrm{d}$) [15], a Population II HADS [16,17,18]. The linear period variation rate $\dot{P}/P$ obtained by refs. [15,19] are both at the order ${10}^{-8}{\mathrm{yr}}^{-1}$.

YZ Bootis (YZ Boo, $\langle V\rangle ={10}^{\mathrm{m}}.57$, $P_0\approx 0.1041\mathrm{d}$) [20], a Population I HADS [21]. In the most recent $O-C$ analysis on YZ Boo [20], the authors obtained a period variation rate of $\dot{P}/P$ = $6.7\left(9\right)\times {10}^{-9}{\mathrm{yr}}^{-1}$. However, they also declared that the second-order polynomial curve fitting does not show sufficient advantage compared to linear fitting, which indicates that it is hard to say whether the period of YZ Boo is varying or not.

GP Andromedae (GP And, $\langle V\rangle ={10}^{\mathrm{m}}.7$, $P_0\approx 0.0787\mathrm{d}$) [22], a Population I HADS [10]. In the latest research of its period variation rate, ref. [22] not only gave a result of $\dot{P}/P$ = $(5.49\pm 0.1)\times {10}^{-8}{\mathrm{yr}}^{-1}$, but also confirmed an amplitude variability in GP And. Two other recent studies for GP And [23,24] showed increasing pulsation period variations with a rate of $\dot{P}/P$∼$6\times {10}^{-8}{\mathrm{yr}}^{-1}$. All these results are much larger than the theoretical values [25].

ZZ Microscopii (ZZ Mic, $\langle V\rangle =9^{\mathrm{m}}.43$, $P_0\approx 0.0672\mathrm{d}$), was first discovered by [26]. In ref. [27], they conducted an $O-C$ analysis on ZZ Mic and obtained that its period was increasing at a constant rate. However, in the latest $O-C$ research of this star [28], the authors claimed that the period of ZZ Mic was increasing at a constant rate during the years 1960 to 2003, and it has been decreasing since 2003.

All of these stars have unsolved problems about their period variations. Thus, we expected to acquire a better understanding on the period variations of them with more data. This paper is organized as follows. In Section 2, we introduce the data sources and the reduction procedures. In Section 3, we show the $O-C$ results of four target stars and their linear period variation rates. In Section 4, we discuss the $O-C$ behaviors and the origins of the period variation rates. In Section 5, we list the conclusions.

2. Data Sources and Data Reduction

The Transiting Exoplanet Survey Satellite (TESS), a NASA mission launched in 2018, is designed to discover exoplanets by detecting transits while simultaneously advancing asteroseismology through its high-precision photometric observations of host stars, enabling scientists to study stellar interiors and oscillations for deeper insights into stellar evolution and structure [29].

In this work, we collected the TML based on the light curves from TESS (from MAST Portal1, which were processed by the TESS Science Processing Operations Center (SPOC) [30,31]), AAVSO2, BAV3 projects, and the historical literature. The overview of the data from TESS, AAVSO, and BAV is listed in

Table 1. Overview of the data from TESS, AAVSO, and BAV.

ID	TESS Sector	Time Spans of AAVSO	Time Span of BAV
XX Cyg	14 (2019), 41 (2021)	2008–2023	2001–2022
YZ Boo	24 (2020), 50 (2022)	2006–2024	2000–2016
GP And	17 (2019)	2008–2023	1998–2014
ZZ Mic	1 (2018), 27 (2020)	2006–2007	—

Note: All the TESS data used in this work have an exposure time of 120 s.

.

For the light curves from TESS, we downloaded the Sectors with an exposure time of 120 s and normalized them using thepython package Lightkurve v2.4.2 [32]. Then, we used the module optimize.curve_fit in thepackage SciPy v1.14.0 [33] to fit the data points around each of the peaks in the light curve with a polynomial to determine the time of maximum light. Due to the profiles of the light curves, we used a third- or fourth-order polynomial, generally, and sometimes a fifth-order polynomial, depending on the quality of the data and the goodness of fit.

Because Julian Days (JD) are used in the AAVSO data, we converted them into Heliocentric Julian Days (HJD) with the help of an online applet4. Moreover, a Monte Carlo simulation was constructed to estimate the uncertainties of each time of maximum light.

For the TML, from the historical literature without uncertainties, we used the following rules to estimate the uncertainties: when the the data value has n decimals, we set the uncertainty as $2\times {10}^{-n}$ (i.e., $0{.}^{\mathrm{d}}002$ for $n=3$, $0{.}^{\mathrm{d}}0002$ for $n=4$). All the TML in this work were converted to BJD with the help of the online applet hjd2bjd5.

3. $\mathbf{O}-\mathbf{C}$ Analysis

All the TML used in this work were collected in a data file, which was uploaded to the Zenodo website (https://zenodo.org/records/14950457, accessed on 1 March 2025). The indicators of the historical literature in the data file are listed in Appendix A.

3.1. $O-C$ Analysis of XX Cyg

For XX Cyg, a total of 905 TML were obtained, of which 360 were determined from TESS, 90 were determined from AAVSO, 36 were determined from BAV, and 419 of them were collected from the historical literature of this star. In order to calculate the $O-C$ values and their corresponding cycle numbers, we adopted the linear ephemeris [15]

$${\mathrm{HJD}}_{\max }=2430671.1102\left(6\right)+0.13486507\left(1\right)\times E,$$

(1)

to obtain a new linear ephemeris

$${\mathrm{BJD}}_{\max }=2430671.102\left(3\right)+0.134865132\left(2\right)\times E.$$

(2)

Then, we fit the $O-C$ values (the residuals of the linear fitting) with a second-order polynomial and obtained the result $O-C=(0.00076\pm 0.00002)+(-4.9\pm 0.1)\times {10}^{-8}\times E+(2.29\pm 0.05)\times {10}^{-13}\times E^2$, which has a linear period variation rate of $\dot{P}/P=(9.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$. The fitting results of the $O-C$ values and corresponding residuals are shown in Figure 1.

3.2. $O-C$ Analysis of YZ Boo

For YZ Boo, a total of 736 TML were obtained, of which 434 were determined from TESS, 28 were determined from AAVSO, 63 were determined from BAV, and 211 were collected from the historical literature. In order to calculate the $O-C$ values and their corresponding cycle numbers, we adopted the linear ephemeris [20]

$${\mathrm{HJD}}_{\max }=2442146.3552\left(2\right)+0.104091579\left(2\right)\times E,$$

(3)

to obtain a new linear ephemeris

$${\mathrm{BJD}}_{\max }=2446624.3756\left(1\right)+0.1040915838\left(8\right)\times E.$$

(4)

Then, we fit the $O-C$ values (the residuals of the linear fitting) with a second-order polynomial and obtained the result $O-C=(-0.00025\pm 0.00001)+(-3.1\pm 0.2)\times {10}^{-9}\times E+(4.8\pm 0.3)\times {10}^{-14}\times E^2$. The obtained linear period variation rate is $\dot{P}/P=(3.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$, which is similar to the result in Ref. [20]. The fitting results of the $O-C$ values and corresponding residuals are shown in Figure 2.

3.3. $O-C$ Analysis of GP And

For GP And, a total of 591 TML were obtained (from 1970 to 2023), of which 268 of were determined from TESS, 33 were determined from AAVSO, 92 were determined from BAV, and 198 of them were collected from the historical literature. In order to calculate the $O-C$ values and their corresponding cycle numbers, we adopted the linear ephemeris [22]

$${\mathrm{HJD}}_{\max }=2441909.4917\left(7\right)+0.07868269\left(8\right)\times E,$$

(5)

to obtain a new linear ephemeris

$${\mathrm{BJD}}_{\max }=2450360.4942\left(1\right)+0.0786828064\left(9\right)\times E.$$

(6)

Then, we fit the $O-C$ values (the residuals of the linear fitting) with a second-order polynomial and obtained the result $O-C=(-0.00214\pm 0.00002)+(-1.106\pm 0.008)\times {10}^{-8}\times E+(3.57\pm 0.03)\times {10}^{-13}\times E^2$.

The obtained linear period variation rate is $\dot{P}/P=(4.22\pm 0.03)\times {10}^{-8}{\mathrm{yr}}^{-1}$, and the fitting results of the $O-C$ values and corresponding residuals are shown in Figure 3.

3.4. $O-C$ Analysis of ZZ Mic

For ZZ Mic, we determined 667 TML from TESS and 2 from AAVSO. Moreover, we also collected 48 TML from the historical literature. Based on the above 717 TML, we determined the new linear ephemeris as

$${\mathrm{BJD}}_{\max }=2449996.6664\left(5\right)+0.0671791806\left(8\right)\times E.$$

(7)

Then, we fit the $O-C$ values (the residuals of the linear fitting) with a second-order polynomial and obtained the result $O-C=(0.00205\pm 0.00002)+(2.37\pm 0.02)\times {10}^{-10}\times E+(-1.27\pm 0.01)\times {10}^{-13}\times E^2$.

The obtained linear period variation rate is $\dot{P}/P=(-2.06\pm 0.02)\times {10}^{-8}{\mathrm{yr}}^{-1}$, and the fitting results of the $O-C$ values and corresponding residuals are shown in Figure 4.

4. Discussion

4.1. Discussion of XX Cyg

The linear period variation rate we obtained in this work was $\dot{P}/P=(9.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$, which is similar to the theoretical evolutionary results in ref. [15]. In this post-mean sequence (post-MS) evolutionary stage, the envelope absorbs energy produced by the hydrogen burning shell and then expands, which leads to an expansion of the star. As it well known, the period of the fundamental mode in HADS mainly depends on the mean density of the star itself. So the increase in the stellar radius means an increase in the period.

In some earlier research (see, e.g., ref. [34]), researchers claimed that XX Cyg experienced a sudden period variation in 1942. However, even though the jumping of period variation is allowed by stellar evolution theory [35], our result indicates that it should be due to the limited data points.

In addition, some other HADS that were suspected to exhibit period jumps have now been proved to have a continuous period variation, or the previously considered ‘jump’ was actually caused by the light-travel time effect (LTTE) (such as CY Aqr [36,37]).

4.2. Discussion of YZ Boo

Different from the other three stars in this work, we found that the second-order polynomial trend in the $O-C$ diagram of YZ Boo was not very significant. Therefore, an F-test was performed to test the advantage of the second-order fitting comparing to the linear one. Based on the routine from Statology6, we used the residuals of the linear and second-order fittings to calculate the p value. It produced $p\approx 0.26$, and then we obtained $1-p\approx 0.74$, which indicates that the goodness-of-fit of the second-order fitting was not significant. As a result, we cannot conclude that YZ Boo exhibited significant period variation in 1955–2024. This result is similar to the conclusion in ref. [20].

However, if we take the linear period variation rate of YZ Boo obtained in this work seriously, it is consistent with the theoretical calculations of the period variation rate in ref. [20], which indicates that YZ Boo is in the post-MS evolutionary stage. In the case of YZ Boo, we need more data to obtain results with higher confidence.

4.3. Discussion of GP And

Although the linear period variation rate of GP And ($\dot{P}/P=(4.22\pm 0.03)\times {10}^{-8}{\mathrm{yr}}^{-1}$) falls into the range of the theoretical prediction (${10}^{-10}{\mathrm{yr}}^{-1}$ to ${10}^{-7}{\mathrm{yr}}^{-1}$) and indicates the star is in the post-MS evolutionary stage, a detailed stellar evolutionary model for this star is also needed to explain the exact value of $\dot{P}/P$ and its evolutionary stage.

Ref. [38] calculated the linear period variation rates of GP And (including both MS and post-MS models) in three cases, and all the results were at the order of ${10}^{-9}{\mathrm{yr}}^{-1}$. Ref. [24] estimated a theoretical value of about $2\times {10}^{-9}{\mathrm{yr}}^{-1}$ from the results in ref. [10]. On the other hand, the observed linear period variation rate of GP And was $\dot{P}/P=(5.49\pm 0.1)\times {10}^{-8}{\mathrm{yr}}^{-1}$ [22] and $5.39\times {10}^{-8}\mathrm{yr}$ [39] in some recent literature.

Why are the theoretical values of the period variation rate of GP And we mentioned above much larger than the observed ones? Here we give two possibilities: (i) First, more accurate and detailed theoretical evolutionary models need to be constructed for GP And, which could give us a more reasonable theoretical prediction of $\dot{P}/P$; (ii) Second, an accumulation of the TML in a longer time-scale is also needed to test whether the observed $\dot{P}/P$ is caused by LTTE. Because a quasi-sinusoidal curve is locally similar to a polynomial curve, the observed second-order polynomial trend could be a superposition of a quasi-sinusoidal trend (from LTTE) and a weak second-order polynomial trend (from stellar evolution) (see, e.g., refs. [7,40]).

4.4. Discussion of ZZ Mic

The situation in ZZ Mic is more complex than that of the other three stars. Historically, some researchers obtained an increasing period [27] and even employed a third-order polynomial to fit the $O-C$ values. The negative linear period variation rate obtained in this work indicates that ZZ Mic has a decreasing period. The different results might come from the lack of TML in cycles from $-100,000$ to 0, which prevents us from obtaining high-confidence results on the period variation of ZZ Mic.

If we ascribe the decreasing period to stellar evolution, it demands that ZZ Mic is in an overall contraction evolutionary stage. In this stage, the star has exhausted all the hydrogen in its core and starts to contract due to gravity. Although some works reported pre-MS stars (which show delta Scuti-type pulsations) can also have negative period variation [41,42,43], the linear period variation rate of ZZ Mic is so small that it can only be a post-MS star [10].

The fitting residuals of the O-C values of ZZ Mic are presented in Figure 5, in which we can find a potential periodic trend. However, the data points were not sufficient to cover a complete cycle. As the TML continues to be accumulated from different observations, we expect to solve out the potential orbital parameters via the $O-C$ values affected by LTTE in this multi-star system (see, e.g., refs. [7,44]) and study some other interesting mechanisms like the magnetic braking effect (which is an important physical mechanism that influences material accretion, spin-down, and evolutionary paths of binary stars [45,46]) in other types of multi-star systems.

5. Conclusions

In this work, we collected TML of four HADS over several decades and obtained the linear period variation rates of these stars via the $O-C$ analysis as follows: (i) XX Cyg, $\dot{P}/P=(9.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$; (ii) YZ Boo, $\dot{P}/P=(3.2\pm 0.2)\times {10}^{-9}{\mathrm{yr}}^{-1}$; (iii) GP And, $\dot{P}/P=(4.22\pm 0.03)\times {10}^{-8}{\mathrm{yr}}^{-1}$; (iv) ZZ Mic, $\dot{P}/P=(-2.06\pm 0.02)\times {10}^{-8}{\mathrm{yr}}^{-1}$.

For XX Cyg and YZ Boo, the period variation rates we obtained are similar to the results in the latest research, which could be ascribed to stellar evolution. For GP And, although the observed period variation rate is in the range of the general theoretical prediction from stellar evolution, some discrepancy appears between the the observed value and some previous specialized studies on this star. More in-depth studies are necessary in the future, based on more observations.

For ZZ Mic, we obtained a negative linear period variation rate, which is different from previous studies and indicates that the star is in an overall contraction evolutionary stage. Moreover, we found a potential periodic (quasi-sinusoidal) trend in the residuals of $O-C$ diagram, which might be caused by LTTE in a multi-star system.

More observations in the future will provide us further opportunity to study these HADS in-depth, including their evolutionary stages and their nature in multi-star systems.