Period modulations of the long secondary periods on the AGB stars

Thanks to long-term ground-based observations such as MACHO (Alcock et al. 1992) and Optical Gravitational Lensing Experiment (OGLE, Udalski et al. 1992), many of the luminous red giants in an evolutionary phase of red-giant branch (RGB) and asymptotic-giant branch (AGB) stars have been known as variable stars. They are called long period variables (LPVs). The pulsating stars are one of the most typical classes of the LPV.

It has also been known that there are period-luminosity (PL) relations between the pulsation periods and stellar luminosities by the studies of the LPVs in the Large (LMC) and Small Magellanic Clouds (SMC) (e.g. Wood et al. 1999; Soszyński et al. 2004, 2011). There are, at least, 6 PL relations attributed to the stellar pulsation periods. In order of increasing the length of the period, the sequences A^′, A, B, C^′, F, and C have been known (Wood et al. 1999; Soszyński et al. 2004; Ita et al. 2004; Tabur et al. 2010; Soszyński & Wood 2013). The sequences C^′, F, and C consist of the Miras and Semi-Regular (SR) variables while the sequences A^′, A, B, and a part of C^′ are the OGLE small amplitude red giants (OSARGs) (e.g. Soszyński et al. 2004; Soszyński et al. 2013). In addition, it has been thought that each PL relation is explained by different pulsation modes (e.g. Takayama et al. 2013; Wood 2015; Trabucchi et al. 2017). The sequence C, the PL relation corresponding to the longest period among the stellar pulsations, can be explained by the radial fundamental mode (Ita et al. 2004; Trabucchi et al. 2021).

On the other hand, the light variations associated with longer periods than the sequence C have been found in such pulsating red giants. The PL relation consisting of those periods is called the sequence D (Wood et al. 1999). Such longer periods are also called the Long Secondary Period (LSP). The length of the LSPs ranges from 400 d to 1500 d which is about 4 times longer than the period in the sequence C in the same luminosity. This indicates that the LSP variations cannot be explained by the radial pulsations on the AGB. Hence there is ongoing debate on the origin of the LSPs.

Here, we review the hypotheses for the LSPs proposed and argued by the previous works. The rotating star with a dark spot is one of the possible explanations for the LSPs. The light curve shapes attributed to the LSP variations were similar to those of the known rotating spotted stars (Soszyński & Udalski 2014). However the rotation periods of a single star in the AGB phase would be longer than the LSPs. Olivier and Wood (2003) found that the rotation velocities of the LSP stars were typically less than 3 km s^-1, which corresponded to the rotation period of $\sim$ 2900 d when an AGB star with R $\sim$ 170 $R_{\odot}$. In addition, Takayama, Wood, and Ita (2015) argued that the theoretical models for an AGB star with a dark spot could not explain the observed colour and magnitude variations involved in the LSP.

The periodic dust formation in the circumstellar space is proposed by Wood et al. (1999). If the circumstellar dusts are formed recurrently, the light from a central star is likely dimmed each time. Wood and Nicholls (2009) found that some LSP stars showed mid-infrared (IR) excess similar to the case of the R Coronae Borealis (RCrB) stars, which implied that the patchy dust clouds might surround the LSP star. On the other hand, Takayama, Wood, and Ita (2015) investigated the colour variations associated with the LSPs among the optical and near-IR bands. They found that the $J$ - $K_{\rm s}$ colours of the oxygen-rich LSP stars barely changed or became bluer when the star dimmed. They also considered the theoretical models assuming a star with a spherical dust shell in a circumstellar space. However, the colour variations derived from the models did not agree with those of the LSP stars.

Non-radial pulsations are one of the possible explanations which have been argued for a long time (e.g. Wood et al. 1999, 2004), and recently the potential pulsation modes were proposed by Saio et al. (2015). They assumed the non-adiabatic and non-radial g^--mode oscillations in a luminous AGB star. In convective conditions, the frequencies of non-radial g^--modes become purely imaginary in adiabatic conditions i.e. non-oscillatory (Shibahashi $\&$ Osaki 1981). But considering strong non-adiabatic conditions, g^--modes can be oscillatory and those oscillation modes are called oscillatory convective modes. Saio et al. found that the oscillatory convective modes can be excited in the convective envelopes of the luminous AGB stars. They also found that the loci of the pulsation periods along the evolutionary trucks of the AGB stars roughly agreed with the location of the sequence D on the PL diagram. In addition, Takayama and Ita (2020) studied the light variations of the LSPs in the optical and near-IR bands. They also assumed a non-radially oscillating star with the dipole modes and found a good agreement with the light amplitudes in the observations.

One of the most likely explanations, along with the non-radial oscillations, for the LSP variations is the comet-like companion, which was first proposed by Wood et al. (1999). This hypothesis assumes eclipsing binaries consisting of a central star and a low-mass companion surrounded by circumstellar dust with a dusty tail (e.g. Wood et al. 1999; Soszyński & Udalski 2014). To estimate the mass of the companion, the radial velocities have been studied. The radial velocity amplitudes associated with the LSP variations lay mainly between 2.0 and 5.0 km s^-1 and the typical value is $\sim$ 3.5 km s^-1 (Hinkle et al. 2002; Nicholls et al. 2009). Assuming a binary system with a total mass of 1.5 M_⊙, the mass of the companion star was 0.09 M_⊙ (Nicholls et al. 2009). They also argued that such binary systems are unlikely because the fraction of 1 M_⊙ main sequence stars with a 0.06–0.12 M_⊙ companion is less than 1$\%$ of the total (i.e. blown dwarf desert). On the other hand, recently Soszyński et al. (2021) studied the mid-IR photometric properties of the LSP variations and found that about half of them showed the secondary minima in their light curves. They argued that the appearance of secondary minima in the light curves is evidence that eclipsing binary is a reasonable explanation for the origin of the LSPs.

Many explanations for the LSP phenomenon have been proposed. Here, we show the advantages and disadvantages of each hypothesis in table LABEL:tab1.

As mentioned above, there have been many attempts to elucidate the origin of the LSPs from the discussion of many properties of the light curves. However, the periodicity of the light variations in the LSP has little understanding. The explanations for the LSP variations would be constrained depending on whether the periodicity of the LSP phenomenon is strong (i.e. regular) or weak (i.e. irregular). For example, the eclipsing binaries including the comet-like companion likely show strong periodicities since the period of the light variations agrees with the orbital period. In other words, if the evidence of the period modulations is found in the light curves associated with the LSP, the eclipsing binaries are inconsistent with the LSP phenomenon. On the other hand, the pulsation periods can change when the mean radius of the star changes. Thus, we perform the wavelet analysis of the optical light curves of the LSP stars and investigate the length of the LSP at cycle-to-cycle. In this paper, we discuss the constraint against the explanations for the LSP proposed in previous works.

We investigate the $I$-band time series obtained by OGLE-II $\&$ -III survey. OGLE-II $\&$ -III are the long-term photometric surveys from 1997 to 2000 (Udalski et al. 1997; Szymański 2005) and 2001 to 2009 (Udalski et al. 2008), respectively, with the $V$ and $I$ bands and observed each star about 500–1000 times with the $I$ band. Soszyński et al. (2009, 2011) obtained 6951 and 1953 LSP candidates in the Large (LMC) and Small Magellanic Clouds (SMC), respectively, from the OGLE database according to the position on the PL diagram. We selected 6904 and 1945 LSP stars in the LMC and SMC, respectively, which have been published in the time series with both the $V$ and $I$ bands within their catalogues. Figure 1 shows the PL diagrams with our sample of LSP stars in the LMC and SMC, respectively, where $W_{I}$ is the Wesenheit index ($W_{I}=V-1.55(V-I)$). We confirmed that the positions of those sample stars were consistent with the location of the sequence D on the PL diagrams.

The observation times of OGLE survey are unequally spaced. In addition, there is annual discontinuation for several month due to its ground-based observations. Thus we adopted the weighted wavelet-Z-transform (WWZ, Foster 1996b) with $I$-band light curves of the OGLE database. The WWZ assumes a Morlet wavelet $F_{\tau,\>\omega}(t)$ for a given time $t$ described as below,

	$\displaystyle F_{\tau,\>\omega}(t)=A_{\tau,\>\omega}$	$\displaystyle+$	$\displaystyle a_{\tau,\>\omega}G_{\tau,\>\omega}(t)\sin[\omega(t-\tau)]$		(1)
		$\displaystyle+$	$\displaystyle b_{\tau,\>\omega}G_{\tau,\>\omega}(t)\cos[\omega(t-\tau)],$

where $\tau$ and $\omega$ are a program time and frequency, respectively. $G_{\tau,\>\omega}(t)$ is a statistical weight with a constant $c$,

$$G_{\tau,\>\omega}(t)=e^{-c\omega^{2}(t-\tau)^{2}}.$$

(2)

As shown above, the Morlet wavelet is the Fourier wavelet with a Gaussian decay profile which means the waves localized near $t=\tau$. The constant $c$ decides the rate of the Gaussian decay against the period $2\pi/\omega$ of the Fourier components. The decay span that the Gaussian decays to 1/$e$ is $1/(\sqrt{c}\omega)$. Hence the number of the periods within the span is $1/(2\pi\sqrt{c})$. We adopt $c=1/(8\pi^{2})$ according to Foster (1996b) i.e. $\sqrt{2}$ cycles are included during the decay span. Adapting the least square fit with the Morlet wavelet $F_{\tau,\>\omega}(t)$ for the light curves, the average value of the wavelet $A_{\tau,\>\omega}$ and the Fourier coefficients $a_{\tau,\>\omega}$ and $b_{\tau,\>\omega}$ are derived.

To investigate the light variations associated with the LSP, the wavelet analysis is performed with a range of $\omega\>\lesssim$ 0.063 d^-1 i.e. $P\>\geqq$ 100 d. We also adopt the range for $\tau$ from $t_{\rm start}+1/(\sqrt{c}\omega)$ to $t_{\rm end}-1/(\sqrt{c}\omega)$ so that the central part of the Morlet wavelet falls enough within the full light-curve interval, where $t_{\rm start}$ and $t_{\rm end}$ are the start and end time of the light curves, respectively. The fit is performed with the intervals of $\delta\tau$=30 d and $\delta\log\omega$=0.01.

Considering the $Z$-statistic of Foster (1996a), we discuss the power spectrum of the WWZ. We define the effective number $N_{\rm eff}$ according to Foster (1996b),

$\displaystyle N_{\rm eff}=\frac{[\sum_{i}G_{\tau,\>\omega}(t_{i})]^{2}}{\sum_{i}[G_{\tau,\>\omega}(t_{i})]^{2}},$

(3)

where $t_{i}$ is $i$ th observation time. And also the weighted variation of the data and the Morlet wavelet are described as below, respectively,

$\displaystyle V_{I}=\frac{\sum_{i}G_{\tau,\>\omega}(t_{i})[I(t_{i})]^{2}}{\sum_{i}G_{\tau,\>\omega}(t_{i})}-\left[\frac{\sum_{i}G_{\tau,\>\omega}(t_{i})I(t_{i})}{\sum_{i}G_{\tau,\>\omega}(t_{i})}\right]^{2},$

(4)

and

$\displaystyle V_{F}=\frac{\sum_{i}G_{\tau,\>\omega}(t_{i})[F_{\tau,\>\omega}(t_{i})]^{2}}{\sum_{i}G_{\tau,\>\omega}(t_{i})}-\left[\frac{\sum_{i}G_{\tau,\>\omega}(t_{i})F_{\tau,\>\omega}(t_{i})}{\sum_{i}G_{\tau,\>\omega}(t_{i})}\right]^{2},$

where $I(t_{i})$ is the observation data of magnitudes. Then the power of the WWZ at ($\tau,\>\omega$) is given as below,

$\displaystyle Z=\frac{(N_{\rm eff}-3)V_{F}}{2(V_{I}-V_{F})}.$

(6)

In the next section, we report the result of the wavelet analysis with the power spectrum and discuss the temporal variations of the length of the LSP.

We investigated the light curves of 6904 and 1945 luminous red-giant variables showing a prominent LSP in the LMC and SMC, respectively. By using the WWZ, we studied the temporal variations of the periods corresponding to the LSPs.

Most power spectrum of our sample of the LSP stars indicated that the period corresponding to the LSPs would be invariant. However, we found that the power spectrum of 101 and 44 LSPs in the LMC and SMC, respectively, showed the signature of the period modulations of the LSPs. There were diversities of the period modulations i.e. monotonic increase or decrease, or constant until the middle and then increase, etc. The comet-like companion is one of the possible explanations, but this hypothesis cannot explain the period modulations because the LSP is determined by the orbital period.

The proportions of the LSP stars showing the period modulations against the whole of our samples in the LMC and SMC were 1.5$\%$ and 2.3$\%$, respectively. Considering TP-AGB stars, the luminosity burst due to the thermal pulse accompanies rapid change of the pulsation periods and it would end in $\sim$ 500 yr, while the quiescent phase after the burst would continue for 10³–10⁵ yr (Vassiliadis $\&$ Wood 1993). Thus, it can be estimated that the AGB stars in the progress of the luminosity burst at a certain moment are about 0.001$\%$–0.1$\%$ of the whole TP-AGB stars. If the LSP variations are associated with the stellar pulsation as proposed by Saio et al. (2015), the period modulations found in this work might be interpreted as the thermal pulse in the AGB stars. Further studies needed for the reasonable explanation of the LSPs.