Inter-Longitude Astronomy project:
long period variable stars
I. L.Andronov
1∗
, V. I.Marsakova
2†
, L. S.Kudashkina
1
, L. L. Chinarova
3
Advances in Astronomy and Space Physics, 4, 3-8 (2014)
© I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova, 2014
1
Department High and Applied Mathematics, Odessa National Maritime University,
Mechnikov str. 34, Odessa, 65029, Ukraine
2
Department of Astronomy, I. I.Mechnikov Odessa National University, Marazliyivska str. 1v, Odessa, 65014, Ukraine
3
Astronomical Observatoty, I. I.Mechnikov Odessa National University, Marazliyivska str. 1v, Odessa, 65014, Ukraine
This article contains the highlights of complex studies of long-period variable stars such as Miras: semiregular,
symbiotic (particularly, pulsating symbiotic), as well as RV Tauri-type stars. In the course of these studies, impor-
tant characteristics of mean light curves were determined. In the case of multi-component variability, additional
periods were found. Correlations between parameters of mean light curves were investigated. The cycle-to-cycle
changes of light curve parameters were analysed using various mathematical methods. Classication criteria of
variable stars and eects of variability were proposed based on this research. The study of observational parame-
ters and the correlations between them can be used to estimate the age, mass, and other physical characteristics of
AGB stars.
Key words: AGB and post-AGB, (stars:) binaries: symbiotic; stars: oscillations (including pulsations); stars:
variables: general
introduction
Studies of long-period variable stars (LPV), as
well as their evolutionary status and pulsating activ-
ity, represent a special interest. These types of stars
are located in the large luminosity and low tempera-
ture region of the H-R diagram. Mira-type stars are
the most numerous. The stars of semiregular types of
variability and OH/IR stars (mainly SR-supergiants)
adjoin directly to them. All these objects  asymp-
totic giant branch (AGB) stars  are intrinsically
related to one another, and are at dierent stages of
their evolution towards the most probable nal stage
of a planetary nebula. Three kinds of circumstellar
maser, discovered from a large number of AGB-stars,
exhibit a particular variation of radiation related to
own rotation. Mira-type stars are studied practically
at all wavelength ranges. This concerns especially
near and far IR-regions. Observations in the visi-
ble spectrum are mainly carried out by astronomy
amateurs.
It is challenging to obtain the light curve of a
Mira-type star because the periods of pulsations are
close to one year and the amplitude is large (5−10m).
Spectral observations are complicated because of the
presence of molecular bands. However, the total vi-
sual light curve is sucient to help form a rst opin-
ion on the star, to classify it and to derive a further
program of its study. At present, several problems
are associated with the study of the Mira-type stars
and (close to them) semiregular stars. The general
problem is to determine the mass.
The majority of Miras are known as single stars.
There are several Mira-type stars in symbiotic sys-
tems (oCet, UVAur, RRTel, RAqr), and the bina-
rity of others is suspected (RAql, UHer). Masses of
Mira-type stars cannot be determined directly from
pulsating theory, since the duration of the pulsat-
ing period varies with time. The radial velocities
undergo a jump, making it impossible to determine
the radius. It is also problematic to use the period
 luminosity relation usually applied for cepheids,
because of the distance uncertainty. So the study of
the observational parameters, which could be used
to estimate masses and ages of the Mira-type stars,
is of a signicant interest. Besides the period of the
brightness variations, the IR-luminosity and maser's
uxes can be used as fundamental characteristics of
a Mira-type star, and these values can be obtained
from observations. All of these parameters are cor-
related among themselves, as well as with the visual
brightness of the star. Perhaps we have a chance to
use these common parameters to divide the Mira-
type stars into two groups: the stars that directly
evolve to white dwarfs, and the stars that pass the
∗
tt_ari@ukr.net
†
v.marsakova@onu.edu.ua
3
Advances in Astronomy and Space Physics I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova
stage of planetary nebulae. Particularly the latter
stars are formed via intensive mass loss at the AGB-
stage.
From studies of the kinematic characteristics it is
known that the Mira-type stars with periods P <
250d and P > 250d belong to dierent stellar pop-
ulations and have dierent initial masses [17]. Our
investigations consider the inter-dependencies of all
photometrical parameters of the light curves of Mira-
type stars (not only the value of the period, which
changes during the Mira-type stage). This makes it
possible to distinguish the stars into groups accord-
ing to the complete set of properties, to estimate
their evolutionary stage relative to one another, and
using the light curves, to mark the groups with dif-
ferent initial masses [22].
Moreover, the shape of the light curve can give
information on the mode of stellar pulsations, if, for
example, it is used in conjunction with the colour in-
dex curves, as is proposed by Dawson & Baird [16],
for 60 objects.
Following the passing of a pulsation-induced
shock wave through the LPV photosphere, the rates
of reactions of the relaxing gas are not always suf-
ciently high to reach the balance conditions prior
to the start of the next pulsating cycle. It can be
reected in the light curve due to dierent eects
(such as change of the maximum amplitude, mini-
mum depth, or duration). Particularly, the hump
on the light curve can be an observable indication
of the shock wave passing [28]. Stars with this pe-
culiar property have a forked structure of their H2O
maser line. If these eects are related, then all stars
that have forked structure of the maser line must
also have a hump on the light curve, even if not in
all cycles of the light variations. It is necessary to
statistically study stars with a clear hump, even if
the hump occurs only in one cycle.
The same issues concern the semiregular vari-
ables. Their investigation is proposed through simi-
lar ways, however at rst it is necessary to separate
all pulsation harmonics, where it is possible, and to
derive the time-dependent relations for their ampli-
tudes and frequencies.
The study of LPVs was carried out as a Stel-
lar Bell part of the international project Inter-
Longitude Astronomy [2] in tight cooperation with
the Ukrainian Virtual Observatory project [43].
atlases of mean light curves
and near infrared curves.
correlation analysis
of mean light curve parameters
For 62 Mira-type stars (in optical band) and 7
Mira-type stars (in near infrared band), the values
of periods have been re-determined; the mean curves
have been tted by a trigonometric polynomial with
an optimal number of harmonics; a number of light
curve parameters have been calculated (amplitude
of the current harmonic in magnitudes, phase of the
amplitude maximum of the current harmonic, ra-
tio of the harmonic amplitude to the amplitude of
the main harmonic wave, the phase shift of the cur-
rent harmonic with respect to the main wave, the
phase shift of the current harmonic with respect to
the maximum brightness). Parameters of the sharp-
ness of the both (ascending and descending) branches
of the light curve are obtained: the average charac-
teristic times (dt/dm) of the brightness changes by
1m; maximal slope of the branch (dm/dt); ratios of
slopes dm/dt to ones derived for the sinusoidal shape
of the light curve with the same amplitude and pe-
riod. Also asymmetry is derived as the ratio of the
ascending branch duration to the period. All these
parameters are tabulated in [7, 23].
To obtain the mean light curves of the Mira-type
stars, we used observations from the databases of the
American Association of Variable Stars Observers
published by Mattei [39] and the data by Whitelock
et al. [46].
We also presented an atlas of the mean light
curves of 34 faint Mira-type stars [25], based on digi-
tized data from the scanned Atlas by Maei & Tosti
1
.
Correlations between the parameters of the light
curves for all these LPV stars are studied. In the pa-
per by Andronov & Kudashkina [6], the correlations
between twenty-ve parameters are discussed.
analysis of variations of
parameters of individual cycles.
correlation analysis of
parameters of individual cycles
As light curves of Miras vary signicantly from
one cycle to another, we did a detailed analysis of
their long-term variations and a correlation analysis
of the parameters of individual cycles of pulsation.
In our sample, approximately 50 Miras, as well as
several semiregular (SRa) variables, were presented.
We used observations from the databases of the
Association of French Observers of Variable Stars
(AFOEV), the Variable Stars Observer League of
Japan (VSOLJ), and recently the American Associ-
ation of Variable Stars observers (AAVSO). This al-
lows to study variability of these stars during a time
interval of up to 100 years. The nal data are dier-
ent for dierent stars, since the databases have been
increasing with time. Since the data are a mixture of
observations from a large number of observers with
individual photometric systems, they show a signi-
cant scatter of 0.3m − 0.5m, although the brightness
1
http://astro.fisica.unipg.it/atlasmaffei/main.htm
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Advances in Astronomy and Space Physics I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova
accuracy estimates conducted by the most experi-
enced observers were considerably better, down to
0.07m. Points that dier from the approximation by
more than 1.2m were removed from the initial data
sets as erroneous. Details of the method of process-
ing the original data can be found in [1, 8].
For each cycle of variability, the following basic
parameters were determined:
ˆ the times of the minimum and maximum, their
phases and magnitudes (two methods were used
to determine the times of the extrema: running
parabolas (RP) and asymptotic parabolas (AP)
as described by Andronov & Marsakova [8]);
ˆ the average time, phase, and magnitude of a
hump at the ascending branch (if it is present);
ˆ the values of inverse slopes dt/dm for indi-
vidual cycles at the ascending and descending
branches;
ˆ the times and phases of the intersections of lin-
ear segments of the light curve with levels of
xed magnitudes.
Details of calculations of these characteristics
were described in the paper [35].
Besides the major parameters, the following re-
lated characteristics were calculated:
ˆ the amplitude of the ascending and descend-
ing branches (dierence in magnitudes at the
current maximum and the two corresponding
minima  the previous and subsequent ones);
ˆ the period, as the time interval between the suc-
cessive maxima;
ˆ the period, as the time interval between the suc-
cessive minima;
ˆ the durations of the ascending and descending
branches;
ˆ the asymmetry (two values: the ratio of the du-
ration of the ascending branch to the period be-
tween nearby maxima and the period between
nearby minima;
ˆ the dierence in magnitudes between the suc-
cessive maxima;
ˆ the dierence in magnitudes between the suc-
cessive minima;
ˆ the average brightness (magnitude) at the as-
cending and descending branches.
We used the characteristics listed above for cor-
relation analysis [35]. Among these correlations, the
most signicant are the correlation between the am-
plitude and the period, as well as between the mag-
nitudes at the maximum and the minimum with the
period. Correlations between magnitudes at current
and previous maxima characterise the stability of the
amplitude and the mean brightness. A correlation
with the opposite sign is observed in VBoo, which is
characterized by amplitude reduction and change in
the character of the variability that is more charac-
teristic of semiregular stars.
Correlations between the magnitudes at current
and next minima also can indicate the presence of
systematic changes in the average brightness.
The time dependencies of all these characteris-
tics were also studied. Here, the time of the current
maximum was ascribed to both periods and asym-
metries, while the average of the time between the
maximum and the corresponding minimum was as-
cribed to such characteristics as the amplitude and
the average brightness, which are encountered twice
in a single cycle.
When necessary, a periodogram analysis was con-
ducted in order to determine the characteristic times
for changes of period, amplitude or mean brightness.
The most prominent changes in the mean bright-
ness are detected for Miras of C spectral class [29].
Carbon variables SCep, UCyg and VCrB show
cyclic changes with characteristic times of approxi-
mately several thousand days. A signicant decrease
of the mean brightness was registered in the car-
bon star VCrB, S-Mira SCas, M-Miras RDra and
RLMi, and may be suspected for some others. Mat-
tei & Foster [40] also detected faintening of many
stars in their sample, and none of the variables be-
came brighter.
We observed a secular increase in amplitude of
5 stars (including 4 Miras and SSVir, which has
been previously classied as a semiregular star) and
a secular decrease in two stars RUCyg and VBoo,
for which the classication as semiregular was con-
rmed. Coecients for the amplitude changes with
time are listed in the paper [35]. We can, there-
fore, assume that some Miras are characterized by a
secular increase of amplitude (except the stars at the
helium ash stage), while some semiregular variables
are characterized by a decrease of amplitude.
The secular (progressive) period changes for 4
stars, which presumably undergo the helium ash,
were studied by Marsakova & Andronov [33]. The
characteristics of these period changes were investi-
gated. TUMi shows a clear increase in the slope
dm/dt of the branches of the light curve, RAql and
RHya show a strong amplitude decrease, mainly due
to brightening of minima. If one places these stars in
the sequence of changes in the amplitude and slope
of the ascending and descending branches of the light
curve, this sequence is in a good agreement with the
position of the stars at dierent stages of the helium
ash on existing models [47].
The other period changes of other Miras will be
described in the next paragraph.
We have also analysed the appearance of the
5
Advances in Astronomy and Space Physics I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova
humps at ascending branches of Miras' light curves
[36]. All the stars that were studied can be di-
vided arbitrarily into two types: rstly, the stars
with large humps, which sometimes appear as sec-
ondary maxima (these stars show humps almost in
each cycle); secondly, the stars with small humps or
steps, which show these features in a relatively low
number of cycles.
As a result, the following conclusions can be
made. The parameters of the humps are related to
the asymmetry of the curve. The relative duration
of a hump correlates especially well with the asym-
metry. This result is in good agreement with paper
[28]. On average, the number of humps increases for
stars with more asymmetric curves. The humps last
longer in low amplitude stars. This correlation is ob-
served over the entire sample of stars. The stars of
the dierent spectral classes (M, C, S) are not dis-
tinguishable according to this dependence, and nor
are the stars which had been previously classied as
semiregular.
Characteristics of individual cycles were pub-
lished in catalogues: for the Mira-type variables by
Marsakova & Andronov [32], [34], for the semiregu-
lar variables by Chinarova & Andronov [12], and for
the RVTau-type stars the catalogue was announced
by Paunzen et al. [42].
Analysis of long-term variability of last men-
tioned types of variables is illustrated by using the
SRb-type variable UDel [4], where two periods of
slow (1198d) and fast variability (119.45d) were re-
ned, and AFCyg [5], which was suggested by Kopal
[18] to belong to the transitional type between the
long-period variables and the RVTau-type stars.
period changes of miras
Many Mira-type variables exhibited changes in
their periods. Zijlstra & Bedding [48] dened con-
tinuously changing periods, sudden changes, and me-
andering Miras (whose periods change with time to
some extent, followed by a return to the previous
period). Our study resulted in a more detailed clas-
sication of the period changes [30, 31].
Small irregular period variations are exhibited by
a majority of the Mira-type variables.
Many Mira variables, for all the time, or just over
certain time intervals, show switchovers between sim-
ilar period lengths producing the saw-tooth O − C
curves. Such evolution is cyclic rather than strongly
periodic with characteristic times of about 10 000
20 000
d
.
For some variables, smooth cyclic period changes
at the timescale of about 1700022000
d
were found.
As it was mentioned above, the progressive pe-
riod changes (continuous ones of the same sign) are
typical for helium shell ash stars.
Some variables have noisy O − C curves as a
result of multiperiodicity eects. They will be dis-
cussed below.
periodogram and wavelet
analyses of the agb
semiregular variable stars
The dierent types of the semiregular variability
using examples of some stars are discussed. Peri-
odogram analysis for nine stars was carried out. The
behaviour of the supergiants PZCas and SPer was
studied in detail, and the radius of SPer was esti-
mated, suggesting that the star pulsates in the rst
overtone [24].
For the stars RXBoo, RTVir, SVPeg, TWPeg,
BKVir, UMon, several values of the periods were
redetermined [27].
Time series analysis of the bright cold carbon SR-
type star YCVn were carried out. The star belongs
to a rare subclass J and has a separate asymmet-
ric envelope. It is assumed that no S process takes
place in this star. Due to this, YCVn may belong
not to the AGB, but to the RGB stage, or to a stage
of helium burning in a nucleus after a helium ash.
The data from the published international databases
of AFOEV (France) and VSOLJ (Japan) were stud-
ied using the periodogram and wavelet analysis and
the running sine approximations. The cycle of vari-
ations is 267d (varying from 247d to 343d), which are
superimposed on 1000
d
10000
d
waves [26].
Thus, the photometric parameters set (ampli-
tude, mean amplitude, amplitudes of the frequencies
completing the multiperiodic oscillation and their
character of the dependence versus time) would be
used for the detailed classication of semiregular
variables [21].
multiperiodicity
Kudashkina [20] noted that the majority of
semiregular variables are multiperiodic. They typ-
ically exhibit two periods with their ratio falling
within the range of 1.70 ≤ P1/P2 ≤ 1.95, although
some examples of stars that are spread even further
apart are known [5].
The group of Mira-type and semiregular variables
with similar periodicity (multiperiodicity) was anal-
ysed. They have periods of 230260
d
and 140150
d
and show intervals of periodical (Mira-type) vari-
ability with a relatively high amplitude and semi-
regular (SR-type) small-amplitude oscillations. Re-
sults of periodogram analysis are represented in [38].
The stars have period ratios within the range of 1.65
1.75, which is in good agreement with [20] and [5].
Four variables were analysed in detail in the paper
[37]. These variables show not only intervals of con-
stancy or a signicant decrease in amplitude (in the
case of RUAnd almost to a zero value [9]), but also
prominent changes in the phase of pulsations. Dur-
ing the semiregular intervals, most of them show
variability aected by two to three periods.
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Advances in Astronomy and Space Physics I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova
symbiotic variables
Symbiotic variables, which contain a compact star
and a red giant, may combine the properties of pul-
sating long-period variables with the activity of in-
teracting binary systems with accretion eects. As
interacting binary systems, they are related to cata-
clysmic variables, and, as red giants with large dust
envelopes, they may be similar to carbon long-period
variables with long-term changes of mean brightness
at timescales of thousands days. Properties of 20
sure and 6 possible symbiotic Miras are discussed
in comparison with the normal Miras by Whitelock
[45].
Chinarova [10] discussed long-term variability of
symbiotic variables UVAur, TXCVn, V1016Cyg
and V1329Cyg. The photographic observations us-
ing the Moscow and Odessa plate collections were
analysed in this research, in addition to the AFOEV
observations. For UVAur and V1329Cyg (which
are not listed by Whitelock [45]) periodic variations
(393
d
and about 1000
d
, respectively) associated with
Mira-type pulsations were revealed. Moreover, for
UVAur, Chinarova [11] detected a long-time wave
(6800
d
) by using a long series of amateur observa-
tions from the AFOEV database. One of the ex-
planations is that the orbital period is close to the
pulsation one, thus the accretion rate may be mod-
ulated also with an orbital-pulsation beat period.
Under these assumptions, the orbital period may be
estimated to be 371.5
d
, but this hypothesis has to be
checked using spectral observations.
For V1329Cyg [15], the main photometrical pe-
riod of 956.5
d
was specied for the post-outburst
state (after explosion as a symbiotic Nova) and the
secondary period (553
d
) was found. A long-term
wave of approximately 5300
d
was suggested.
For V1016Cyg, the pulsational period of 450
d
is
listed by Whitelock [44]. The 5500
d
periodicity was
interpreted as corresponding to the orbital period
[41]. The ares seen with this period are similarly
explained by the enhanced mass transfer at the pe-
riastron of eccentric orbit.
In the classical symbiotic variable CHCyg [3],
three components of oscillations were studied: the
long-term (1840
d
), orbital (694
d
) and probably pul-
sation (99.6
d
) ones.
For the symbiotic Mira RAqr, the period of pul-
sations was specied (387.51
d
) and also secondary
cyclicity of about 3955
d
was detected [13]. The time
scale ranges from 7.1 to 34.1 days/mag at the ascend-
ing branch, to 25.744.6 days/mag at the descending
branch.
conclusion
Evidence exists that the physical processes tak-
ing place in the star during the observation interval,
are reected on its light curve. However, the mean
light curve of the star undergoes the characteristic
evolutionary variations (the amplitude is changed,
the period and asymmetry are increased, the humps
appear at the light curve).
At the present time, the AGB-stage is subdi-
vided into the earlier (EAGB) and thermally pulsing
regime (TPAGB) of the He-burning shell [14]. At
the TPAGB stage luminosity of the star increases
(soMbol decreases). For example, S-stars correspond
to the EAGB according to the range of the lumi-
nosity and eective temperatures. Thus, one may
try to classify the stars using all obtained parame-
ters, determined from the light curve, without use of
the relationships between the absolute and apparent
magnitude.
Thus, the observational parameters and the cor-
relations between them could be used to estimate
the age, mass and other physical characteristics for
the AGB stars. Especially interesting is to carry out
the study of stability of the mean parameters of the
light curves of the Mira-type stars; to estimate the
contribution of every harmonic wave, contributing
to the total light curve; to determine the role of the
hump at the total description of the light curves; to
investigate evolution of the hump at the light curve
for individual stars; to study the similarity of the
photometric characteristics in the SR-type stars and
Mira-type stars, bringing them to the physical like-
ness.
Our research allows to determine various proper-
ties of long period variables stars and dependencies
between their parameters. Also we propose dierent
ways to classify them. For example, we can use as the
criteria the types of long-term period changes, sec-
ular amplitude changes, statistical diagrams based
on deviations of variability characteristics from one
cycle to another as described by Marsakova & An-
dronov [35] and Marsakova [30].
Also, we oer additional photometric classica-
tion of Mira-type stars based on three groups of pa-
rameters of the mean light curve: total P , ∆m, f
 period, amplitude, asymmetry; additional rk, ϕk,
s  wave amplitude curve with frequency k · f1,
the phase shift relative to the main wave, the de-
gree of the trigonometric polynomial, approximat-
ing light curve and the parameters of the slope
mi, ti,mis,md, td,mds  the slope of the ascending
branch, the increase of brightness at 1m, the dier-
ence between the ascending branches from the corre-
sponding branch of the sine wave (index i), the same
for the descending branch (index d) [19, 23].
acknowledgement
We would like to thank all amateur astronomers
from AFOEV, VSOLJ and AAVSO, who made this
research possible.
7
Advances in Astronomy and Space Physics I. L.Andronov, V. I.Marsakova, L. S.Kudashkina, L. L. Chinarova
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