SAG: Stars and Galaxies Vol. 6, id. 6, 2023 December 28
c 2023. Center for Astronomy, University of Hyogo.
⃝

Period modulations of the long secondary periods on the AGB stars
Masaki Takayama1∗
1

Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo,
407–2 Nishigaichi, Sayo-cho, Hyogo 679–5313, Japan∗
takayama@nhao.jp(Received 2023 October 31; accepted 2023 December 5)

Abstract
30%–50% of the luminous pulsating red-giant stars show light variations of a longer period than the pulsation
periods. Those periods are called long secondary periods (LSP). There has been debated for many years but theorigin of the LSP is still unknown. To explain the LSP variations, there have been many approaches in not only
observations but also theoretical studies. However, the invariance of the length of the LSPs has been investigatedlittle. Thus, we studied the temporal variations of the period by performing the weighted wavelet-Z-transform
analysis.
Using the OGLE-III database, the I-band light curves of 6904 and 1945 LSP candidates in the Large/Small
Magellanic Clouds, respectively, were analyzed. Most of our sample stars indicated that the period corresponding
to the LSP was constant during the observation term. However, 101 and 44 LSP stars in the LMC and SMC,
respectively, showed the signature of the temporal variations 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-likecompanion is one of the possible explanations for the LSP variations, but this hypothesis cannot explain the period
modulations because the LSP is determined by the orbital period.
Key words: infrared: stars – stars: AGB and post-AGB – stars: late-type
1. Introduction
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 ofthe luminous red giants in an evolutionary phase of red-giant
branch (RGB) and asymptotic-giant branch (AGB) stars havebeen known as variable stars. They are called long period variables (LPVs). The pulsating stars are one of the most typicalclasses 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 lengthof 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 ofC′ 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, canbe 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 arealso 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 theradial pulsations on the AGB. Hence there is ongoing debate
on the origin of the LSPs.
Here, we review the hypotheses for the LSPs proposed andargued 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 & Udalski2014). 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 typicallyless than 3 km s−1 , which corresponded to the rotation periodof ∼ 2900 d when an AGB star with R ∼ 170 R⊙ . In addition,
Takayama, Wood, and Ita (2015) argued that the theoreticalmodels for an AGB star with a dark spot could not explain theobserved 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2

M. Takayama

LSP stars showed mid-infrared (IR) excess similar to the caseof the R Coronae Borealis (RCrB) stars, which implied thatthe patchy dust clouds might surround the LSP star. On theother hand, Takayama, Wood, and Ita (2015) investigated thecolour variations associated with the LSPs among the opticaland near-IR bands. They found that the J - Ks colours of theoxygen-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-adiabaticand 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 consideringstrong non-adiabatic conditions, g− -modes can be oscillatory
and those oscillation modes are called oscillatory convectivemodes. 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 theobservations.
One of the most likely explanations, along with the nonradial oscillations, for the LSP variations is the comet-likecompanion, 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, theradial 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 ∼ 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 theother hand, recently Soszyński et al. (2021) studied the midIR photometric properties of the LSP variations and found thatabout 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 ofeach hypothesis in table 1.
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 lightVol. 6,variations in the LSP has little understanding. The explanations for the LSP variations would be constrained depending onwhether 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 withthe 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 performthe wavelet analysis of the optical light curves of the LSP stars
and investigate the length of the LSP at cycle-to-cycle. In thispaper, we discuss the constraint against the explanations for the
LSP proposed in previous works.
2. Data
We investigate the I-band time series obtained by OGLE-II& -III survey. OGLE-II & -III are the long-term photometricsurveys from 1997 to 2000 (Udalski et al. 1997; Szymański2005) 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 withboth 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 WI is the Wesenheit index (WI =V − 1.55(V − I)). We conﬁrmed that the positions of thosesample stars were consistent with the location of the sequence
D on the PL diagrams.
3. Analysis
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 Fτ, ω (t) for a given time t described as below,Fτ, ω (t) = Aτ, ω + aτ, ω Gτ, ω (t) sin[ω(t − τ )]+ bτ, ω Gτ, ω (t) cos[ω(t − τ )],(1)

where τ and ω are a program time and frequency, respectively.
Gτ, ω (t) is a statistical weight with a constant c,
Gτ, ω (t) = e−cω (t−τ ) .
2

2

(2)

As shown above, the Morlet wavelet is the Fourier wavelet witha Gaussian decay proﬁle which means the waves localized neart = τ . The constant c decides the rate of the Gaussian decay
against the period 2π/ω of the Fourier components.
The decay√
span that the Gaussian decays to 1/e is 1/( cω).
√ Hence thenumber of the periods within the span is 1/(2π √c). We adopt
c = 1/(8π 2 ) according to Foster (1996b) i.e. 2 cycles areid. 6

Period modulations of the LSPs

3

Table 1. The hypotheses proposed by previous works and the results of observations for the LSPs.

Near-IR

Secondary minima

colour variationsin the light curves

Uncertain

DisagreeDisagreeUncertain

Uncertain

DisagreeDisagree

Agree

DisagreeUncertain

Uncertain

Uncertain

Agree

Agree

Agree

Agree

Uncertain

Agree

Agree

Uncertain

Uncertain

Agree

Periodic

Lengthlight variationsof the LSP

AgreeDisagreeCircumstellar dust

Uncertain

Radial pulsations

Hypothesis
Rotating star

PL relation

with a dark spot

Non-radialpulsations(g− modes)
Comet like

11

11

12

12
WI=I-1.55(V-I)

WI=I-1.55(V-I)

companion

13

14

15

13

14

15

16

16
1

1.5

2

2.5

3

1

1.5

2

logP

2.5

3

logP

Fig. 1. [Left] Period vs WI diagram with the sample of the LMC LSP candidates. [Right] same as the left panel but of the SMC LSP candidates.

included during the decay span. Adapting the least square ﬁt
with the Morlet wavelet Fτ, ω (t) for the light curves, the average value of the wavelet Aτ, ω and the Fourier coefﬁcients aτ, ωand bτ, ω are derived.
To investigate the light variations associated with the LSP,
the wavelet analysis is performed with a range of ω <∼ 0.063d−1 i.e. P √>= 100 d. We also√adopt the range for τ fromtstart + 1/( cω) to tend − 1/( cω) so that the central partof the Morlet wavelet falls enough within the full light-curveinterval, where tstart and tend are the start and end time of thelight curves, respectively. The ﬁt is performed with the intervals of δτ =30 d and δ log ω=0.01.
Considering the Z-statistic of Foster (1996a), we discuss thepower spectrum of the WWZ. We deﬁne the effective number
Neﬀ according to Foster (1996b),∑[ Gτ, ω (ti )]2
Neﬀ = ∑ i,
(3)
2
i [Gτ, ω (ti )]where ti is i th observation time. And also the weighted variation of the data and the Morlet wavelet are described as below,

respectively,∑[∑]2Gτ, ω (ti )[I(ti )]2i∑
i Gτ, ω (ti )I(ti )
∑
VI =
−
,
i Gτ, ω (ti )i Gτ, ω (ti )
and

∑

VF =

2
τ, ω (ti )[Fτ, ω (ti )]i G∑i Gτ, ω (ti )

[∑
−

(4)

τ, ω (ti )Fτ, ω (ti )
iG
∑

]2
,

i Gτ, ω (ti )(5)where I(ti ) is the observation data of magnitudes. Then the
power of the WWZ at (τ, ω) is given as below,
Z=(Neﬀ − 3)VF.
2(VI − VF )(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.

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M. Takayama

Vol. 6,

OGLE-LMC-LPV-06844

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Fig. 2. [Left column]The light curves (upper) and contour map (lower) of the power spectrum of OGLE-LMC-LPV-06844. The observation errors in
the I band on the light curves are typically smaller than about 0.01 magnitude. The colours in the contour map show the power corresponding to Zof the WWZ. The horizontal dotted magenta line shows the period published by the OGLE project. [Right column]Same as the left column but of
OGLE-LMC-LPV-10956.

4. Results and discussion
4.1. Light curves and power spectrum
Figure 2 shows samples of the light curves and contour mapsof the power spectrum derived from the WWZ. Most of oursample LSPs showed a good agreement of the period publishedby the OGLE with the period corresponding to the peak of thepower spectrum. As mentioned in section 3, the width of thecentral part of the Morlet wavelet depends on the program frequency ω (i.e. period). Thus, the range of the program timeτ which our wavelet analysis can perform becomes shorter ina longer program period. This is the reason why the coloured
region on the contour map is triangle shape. It also means that
our method
can only performed when the length of the LSP is√
1/(2 2) times shorter than the time span of the observations.
4.2. Case A: regular light variations
Although there were diversities of the pattern of the contour
map of both the LMC/SMC stars, the most typical contour map
shows the power spectrum in which the period corresponding
to the peak is almost constant. Two contour maps of ﬁgure 2
correspond to the examples of the contour maps with horizontal
ridges. The light curves of those samples are typically regularand show prominent light variations associated with the LSP.
As shown in the left-bottom panel of ﬁgure 2, OGLE-LMCLPV-06844 shows a power spectrum with a continuous ridgeon the contour map. On the other hand, we also found thatsome contour maps show multi-humps or discrete ridges along
the horizontal axis. One of the explanations for such detachedridges is the amplitude variations in the light curves. The
right column of ﬁgure 2 is the light curves and contour mapof OGLE-LMC-LPV-10956. We can see variations in the light
amplitudes. The regions corresponding to larger amplitudes
among the light curves were consistent with the location of theridges corresponding to large power. However, in order to consider other possible explanations, further research is needed.
Another case of the contour map was a slightly meandering ridge as shown in the left-bottom panel of ﬁgure 3. We
can see a ridge meandering around the horizontal dotted line.
This might indicate period variations of LSP but it is unlikely
by analogy of eclipse and ellipsoidal binaries. We studied
light curves of the red-giant binaries in the SMC obtained by
Soszyński et al. (2011). The right-bottom panel of ﬁgure 3
shows the contour map of a sample of the red-giant binaries
showing a meandering ridge. The periods of the light variations caused by those eclipse/ellipsoidal binaries should beconsistent with the orbital periods and, thus generally are invariant. Further studies are needed to explain the reason for the
meandering ridges found in the eclipse/ellipsoidal binaries. On
the other hand, at least, the amounts of the period variations
larger than the meandering obtained in the eclipse/ellipsoidal
binaries would be required for the evidence of the modulationsof the LSPs.

id. 6

Period modulations of the LSPs
OGLE-SMC-LPV-04470

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Fig. 3. Same as ﬁgure 2 but of [left] OGLE-SMC-LPV-04470 and [right] a sample of the LPVs showing the light variations by binarity(OGLE-SMC-LPV-10486).

4.3. Case B: period modulation
As shown above, we studied the light curves and power spectrum of all our samples by eyes. The power spectrum of most of
our samples suggested that the period associated with the LSPs
was constant or at least barely changed during the observation
term. Although some of those spectra showed ﬂuctuation of
the power, the period corresponding to the peak on the contour
map was almost invariant.
However, some power spectra seemed to indicate a strong
temporal change according to the contour map. This implies
the period modulations. The features of those ridges on thecontour maps were diverse and complex so it was difﬁcult togive criteria in a quantitative way. Hence we set a simple ruleto determine by eyes the candidates showing the period modulations as below.
The periods corresponding to the LSP have changed assimilar to or larger than about the full-width of half maximum of the ridge on the contour map even once.
According to the above criteria, we found that the sample of
101 and 44 LSPs in the LMC and SMC, respectively, showedsigniﬁcant changes in the period corresponding to the peak of
the power spectrum. The ratios of the number of possible candidates against the whole of our sample were 1.5% and 2.3%
in the LMC and SMC, respectively.
The temporal changes of the periods corresponding to the
LSP variations would give us a strong limitation for the explanations for the LSP. In the comet-like companion, the lengthof the LSP should be consistent with the orbital period which
would be constant. This hypothesis needs to reconsider how to
explain the change in the orbital period.
On the other hand, the pulsation periods in luminous AGB
stars have theoretically been predicted to change the length during the cycle of the thermal pulse on the AGB star. Vassiliadis
and Wood (1993) considered Mira-type stars in the evolution
phase of thermally pulsating AGB (TP-AGB) stars. Once athermal pulse occurs, the stellar surface luminosities rapidly
increase and decrease like a “spike”, then slowly recover for along time. The pulsation periods corresponding to the radial
fundamental modes also showed rapid change in a short timeand slow recovery. Those results indicate that the effective temperatures and stellar radii also change depending on the surface
luminosities.
The intervals of the cycle-to-cycle of the thermal pulse were
103 –105 yr, depending on the stellar masses and the chemicalcompositions, while the spike was only ∼ 500 yr. Thus, theratios of the timescale corresponding to the short burst and thequiescent phase are ∼ 10−3 –10−1 . As mentioned above, theproportions of the LSP stars showing the period modulations
were 1.5% and 2.3% in the LMC and SMC, respectively. If
the origin of the LSP is stellar pulsations, the LSP stars showing the period modulations might be interpreted as the thermal
pulse occurring. Further studies need to conclude.
Figure 4 shows the example of the power spectrum withprominent temporal variations suggesting period modulationsin the LSP variations. In most cases, the periods corresponding

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M. Takayama

Vol. 6,

OGLE-LMC-LPV-00391

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Fig. 4. Same as ﬁgure 2 but of [left-top] OGLE-LMC-LPV-00391, [right-top] OGLE-SMC-LPV-07849, [left-bottom] OGLE-LMC-LPV-13964, and[right-bottom] OGLE-SMC-LPV-14466, respectively.

id. 6

Period modulations of the LSPs
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HJD-2450000[d]Fig. 5. [Left] the time series of the magnitudes in the I, J, H, and Ks bands of OGLE-SMC-LPV-14466. [Right] the time series of the J − Ks andH − Ks colours of OGLE-SMC-LPV-14466.

12.2

Ks

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Fig. 6. The
J − KsvsKs
diagramplottedwith
OGLE-SMC-LPV-14466. The black ﬁlled circles are the locusof the observations from HJD = 2,452,212.5 d to HJD = 2,458,083.3
d with SIRIUS camera. The red solid lines correspond to thelocus near the large dimming (HJD = 2,452,395.6–2,452,514.4).

to the peak of the power spectrum monotonically increased or
decreased during the observation term.
On the other hand, the contour map of OGLE-LMC-LPV13964 showed the meandering ridge as similar to OGLE-SMC-

LPV-04470 shown in ﬁgure 3 . However the ridge corresponding to the pulsation periods of 170.82 d and 152.05 d publishedby OGLE also meandered in synchrony with the ridge corresponding to the LSPs. This might indicate that the length of
the LSPs correlates with the stellar radii since the pulsation
periods depend on the stellar radii and masses (i.e. the meandensities of a star). Further study is needed for the correlations
between the pulsation periods and the LSPs.
Another notable object is OGLE-SMC-LPV-14466. Only
OGLE-SMC-LPV-14466 showed that the periods corresponding to the LSP variations suddenly started to increase. The
right-bottom panel of ﬁgure 4 shows the light curves and power
spectrum obtained from OGLE-SMC-LPV-14466. As you cansee, the ridge corresponding to the LSPs is partly horizontal
on the contour map, which indicates that the periods of the
LSP variations were almost constant until near the HJD of
2,452,500. On the other hand, during the later term of theobservations, the ridge suggests a continuous increase in thelength of the LSP.
We also notice that according to the light curves, the large
dimming occurred in OGLE-SMC-LPV-14466 near HJD of
2,452,480. To study the properties of the large dimming, we
combined the OGLE I-band magnitudes with the light variations in the near-IR bands obtained with the SIRIUS camera in
the IRSF 1.4 m telescope (Ita et al. 2018). The left panel ofﬁgure 5 shows the I-, J-, H-, and Ks -band magnitudes near

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M. Takayama

the large dimming in the I band. As you can see, the feature of
the large dimming has been obtained from not only the I-band
light curves but also the near-IR band light curves. The rightpanel of ﬁgure 5 shows the colour variations of the J − Ks andH − Ks . Although the light curves showed considerable dimming in both the optical and near-IR bands, the colours at thetime were not noticeably redder than the colours at the normal
maxima. We also show the locus of OGLE-SMC-LPV-14466
on the J − Ks vs Ks diagram obtained by the SIRIUS camera in ﬁgure 6. The locus corresponding to the large dimming
disagreed with the locus of the normal LSP variations. Those
results would indicate that the large dimming was unlikely to
be involved by the LSP variations of the star. However, thepower spectrum obtained by the WWZ suggests that just after
the large dimming occurred, the LSP started to increase. Thus,
the trigger of both the increase in the length of the LSP and thelarge dimming might be the same physics.
5. Summary
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 thesignature of the period modulations of the LSPs. There were
diversities of the period modulations i.e. monotonic increaseor 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∼ 500 yr, while the quiescent phase after the burst would continue for 103 –105 yr (Vassiliadis & Wood 1993). Thus, it canbe estimated that the AGB stars in the progress of the luminosity burst at a certain moment are about 0.001%–0.1% of thewhole TP-AGB stars. If the LSP variations are associated withthe stellar pulsation as proposed by Saio et al. (2015), the period modulations found in this work might be interpreted as thethermal pulse in the AGB stars. Further studies needed for thereasonable explanation of the LSPs.
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