SAG: Stars and Galaxies Vol. 6, id. 11, 2023 December 28
c 2023. Center for Astronomy, University of Hyogo.
⃝Linear Nonadiabatic Pulsation Models of Ultra-Long-Period Cepheids
Toshihito Ishida 1∗
1

Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo,
407–2 Nishigaichi, Sayo-cho, Hyogo 679–5313, Japan∗ishida@nhao.jp(Received 2023 October 31; accepted 2023 December 20)

Abstract
Ultra-Long-Period Cepheids (Periods longer than 80 days) are examined using linear nonadiabatic pulsation
models. We found models with ultra-long-periods even with Z = 0.02 models, so such a Cepheid may exist in ourGalaxy, which has not been discovered yet. We seem to need a survey of the far side of our Galaxy or a survey in theother galaxy with a similar chemical composition to discover such an object. It is also shown that the models withperiods longer than 200 days need to be more massive than the Cepheid upper mass limit, inferred from the stellar
evolution theory. It is pointed out that the Cepheid’s nature of HV 1956 in SMC, which has over-200 days period,
needs careful conﬁrmation.
Key words: Stars: individual (HV 1956 in SMC) — stars: oscillations — stars: variables: Cepheids
1. Introduction
Classical Cepheids are radially pulsating population I yellow
supergiants. These stars are known to be the standard candles
for the extragalactic distance scale, using the Leavitt law, previously called the period-luminosity relation. Their fundamental
properties are explained by the stellar pulsation theory. It isfound that there are some differences in pulsation properties
among the stellar systems of different chemical compositions,
including the period-luminosity relation. Therefore, it is interesting to investigate the difference between pulsation models
with different chemical composition parameters.
As one of the possible results of the difference in chemicalcomposition, it is known that the longest period of Cepheids in
our Galaxy, the Large Magellanic Cloud (LMC), and the Small
Magellanic Cloud (SMC) have large differences. While thestar with the longest period among the classical Cepheids in
our Galaxy is S Vul with ∼ 68 days, the Cepheids with periods
longer than 100 days have been known in both the LMC and theSMC. Among these stars, the Cepheid with the longest period
is the HV 1956 with P = 208.7992166 days 1 However, it isunclear whether the differences in the periods of the Cepheids
between the LMC/SMC and our Galaxy are due to the differences in the chemical compositions, or not, because a large
portion of the Cepheids in our Galaxy have not been discovered yet due to the heavy extinction to galactic-disk direction.
Such Cepheids or Cepheid-like variables with long periods andhigh luminosities have a special advantage in distance measurement. Grieve, Madore, and Welch (1985) suggested thatsome bright stars (called by them as “Leavitt Variables”) may
have the period corresponding to the extension of the Cepheid’s
1Period of the HV 1956 is from the OGLE Collection of Variable Stars site.
https://ogledb.astrouw.edu.pl/ ogle/OCVS/

period-luminosity relation. Recently, such a possibility with
”Ultra Long Period”(ULP) Cepheids has been re-investigatedby some authors like Bird, Stanek, and Prieto (2009). TheULP Cepheids are deﬁned by them as the fundamental-mode
Cepheids with a pulsation period of longer than 80 days. Morerecent investigations concerning the ULP Cepheids are summarized in Musella (2022).
There are a few investigations on pulsation models ofCepheids with a long period. Carson and Stothers (1984) studied the Cepheids with periods of longer than 50 days using
the dynamic models, but it was before the OPAL opacities appeared. Convective models with periods over 100 days are reported in Fiorentina et al. (2007). However, because only a few
models have periods correspond to the ULP Cepheids, it seems
difﬁcult to discuss the properties of the ULP Cepheids fromthe studies of pulsation models. A part of the linear nonadiabatic calculations of the long-period Cepheids are previously
reported in Ishida (1998).
In this paper, I would like to examine the existence of thepulsation modes with periods over 100 days and the pulsation
stability of the modes using the linear nonadiabatic pulsation
models. The evolution model of such a star is not calculated
in this paper. I also would like to discuss the Cepheid’s natureof the HV 1956. In section 2, the models we constructed areexplained. The results are shown in section 3. The discussions
are presented in section 4. We summarize in section 5.
2. Models
For the calculation of the linear non-adiabatic pulsation periods, we used the same code used in Ishida (1995), which uses
Castor’s (1971) type procedure to estimate the periods and thegrowth rates in the non-adiabatic pulsations using the OPAL
opacity tables (Iglesias et al. 1992; Rogers & Iglesias 1992).

2T. Ishida

As usual for radial stellar pulsation in Cepheids, only the stellar envelope is included in the basic equations, since the pulsation amplitude is very small near the stellar core. Accordingly,thermonuclear energy production is neglected and constant luminosity ﬂux is assumed at the inner boundary. The effect ofconvection is not included in this model, and the envelope istreated as purely radiative. We can calculate pulsation periods
and growth rates for radial stellar pulsation modes up to thearbitrary order with this code.
The effective temperature of the models is ﬁxed to 5000 K.
The effective temperatures of the long-period Cepheids arelower than those of short-period Cepheids like the doubleperiod Cepheids. Therefore we need to assume the effective
temperature values of lower than those values used in the models for the short-period Cepheids (Ishida 2017). Molecularopacities are not included in the opacity tables used.
Considering the result of preliminary calculations, the model
mass range is set from 10.0 to 40.0 M te with mass step
2.0 M te. We increased the luminosity of the models fromt until the model does not converge with the lumi10000.0 L enosity step 10000.0 L te. For a very high luminosity model,
we can calculate the static-model envelope but we cannot obtain the periods of the linear-adiabatic pulsations for no convergence. Because we usually experience the no convergenceerror in the nonadiabatic part if there is a problem in the numerical calculation, such an error in the adiabatic part seems toindicate that pulsation like Cepheids is not possible in such aregion. We will further discuss this no convergence of the codein section 4.
Since the ULP Cepheids are observed in many systems,
we calculated models with three different compositions, say,Z = 0.02, 0.01, and 0.004.
3. Results
The result for Z = 0.02 is shown in ﬁgure 1. Constant period lines for 50, 80, 100, 150, 175, 200, and 210 days areindicated by black lines with its fundamental periods indicatedaside. Because the ULP Cepheids are deﬁned as Cepheids witha period of longer than 80 days, it is worth noting that themodel with a period of longer than 80 days is widely available for the mass range we investigated. The red line indicatesno convergence limit. As the mass of the model increases, wecan obtain the period for the models with higher luminosities.
The longest period we can obtain also increases as the model
mass increases. It is worth noting that the period over 200 days
becomes possible over 24 M te in Z = 0.02 models. The fundamental mode is pulsationally unstable in all the models wecalculated.
Figures 2 and 3 are results for Z = 0.01 and Z = 0.004,respectively. It is shown that the no-convergence line andconstant-period lines shift to the higher luminosity direction
as the values of Z decrease. However, the longest possible period does not extend to the lower mass region, and we still needmore than around 25 M te to obtain the model with a period ofover 200 days.
We also summarized in Figure 4, to clarify that the longest
period increases as the model mass increases. The longest period reaches 200 days around 24 M te for Z = 0.02, and aroundVol. 6,Fig. 1. Constant period plot for Z = 0.02. The abscissa is the model
mass and the ordinate is the model luminosity both in the solar unit.
Black curves are constant-period lines for 50, 80, 100, 150, 175, 200,and 210 days with its periods indicated aside. The red line indicates noconvergence limit.
t for Z = 0.01 and Z = 0.004 models.
27 M e4. Discussions
For evaluating the effect of convection and inclusion ofmolecular opacities, we tried to use the Radial Stellar Pulsation(RSP) feature of the Modules for Experiments in Stellar
Astrophysics (MESA) project, which is described in the MESA
paper5(Paxton et al. 2019). We used almost the same parameters as the default values used in the MESA, however, grid
points are increased to nearly 500, as our no-convection model
also uses nearly 500 zones. We picked up only the results forthe linear calculation part of the MESA RSP feature. The mostsigniﬁcant difference between the results of MESA RSP andour models is that the MESA models do not converge at thelower luminosity than our radiative models, thus we cannot
construct models with a long period. The converged model
has periods of about 10% difference and smaller growth rates
than the results obtained by the radiative models with the same
mass and luminosity, therefore the main results with our models seem not changed by the inclusion of convective energy
transport. We have not found a set of model parameters tobring a period consistent with the ULP Cepheids’ periods byusing the MESA feature. Because the calculated period for theRV Tauri models with the MESA are also reported not to reachid. 11

Linear Nonadiabatic Pulsation Models of the Ultra-Long-Period Cepheids

Fig. 2. Constant period plot for Z = 0.01. Notations are similar to ﬁgure 1.

a period of longer than 20 days, the MESA RSP feature seems
to need modiﬁcation to extend to the longer period region.
As mentioned in Section 2, our code does not converge at
the luminous end of the ﬁxed mass sequence. In most case,the linear-adiabatic part does not converge. We have examined the acoustic cut-off frequency (for example, Unno et al.
1989) proﬁle from the static model and compared it with theperiod obtained. For the models with a period of longer than
200 days, the period in the linear-adiabatic pulsations is about
230 days while the corresponding period for the maximum of
the acoustic cut-off frequency is around 13 days. Therefore, noconvergence does not seem from this physical limitation. We
have not identiﬁed the origin of no convergence for now.
The ULP Cepheids are possible even in Z = 0.02 models, as
described in the previous section. Although we have not found
Cepheids with a period of longer than 80 days in our Galaxy,
we have obtained the envelope model up to 150 days even at
10.0 M te. The ULP Cepheid in our Galaxy may be overlooked
for the strong distinction to the Galactic disc direction. We
seem to need a survey in the far side over the Galactic center
to ﬁnd such an object. It may also be possible to ﬁnd such an
object in other stellar systems with similar compositions likeour Galaxy.
We also pointed out that periods over 200 days are possiblearound 24 M te or more massive regions. Cepheids are usually considered to be observed in the so-called blue-loop phaseof the stellar evolution. The upper limit of the initial massthat the model star indicates this evolutionary phase depends

Fig. 3. Constant period plot for Z = 0.004. Notations are similar toﬁgure 1.

Fig. 4. The longest periods obtained as the model mass changes. The
black circles are for Z = 0.02 models. The red triangles and blue
squares are for Z = 0.01 and Z = 0.004 models, respectively.

34T. Ishidaon details included in the evolution calculation. However, forexample, Anderson et al. (2014) estimated the upper mass
limit for Cepheids as around 12 M te. Although HV 1956
in SMC has been treated as a classical Cepheid since Shapley
and MaKibben Nail (1951), conﬁrmation of its properties as aCepheid seems to be needed.
5. Concluding Remarks
In this paper, we calculated linear nonadiabatic pulsation periods with the ULP Cepheids models. The ULP Cepheids seem
possible even with Z = 0.02 models, therefore there is a possibility that such a long-period Cepheid may exist somewherein our Galaxy. We seem to need a survey of the far side of ourGalaxy or a survey of the other galaxies with similar chemicalcomposition.
It is also shown that the models with a period of longer than
200 days need to be more massive than the usual Cepheid upper mass limit. So, although the estimated magnitude may benearly on the extension of the Cepheid period-luminosity relation, the variables with over 200 days period may be evolutionary different from classical Cepheids. Especially, it is pointed
out that HV 1956 in SMC with a period of longer than 200days, which is usually considered as a classical Cepheid, needs
careful conﬁrmation on its nature as a Cepheid.
Acknowledgments
I would like to thank my colleagues for their useful comments during this investigation. Computations are carried out
mainly on the workstation provided by a research grant fromthe University of Hyogo.
I am also indebted to the anonymous referee whose comments improved this manuscript, especially on the discussion
concerning the acoustic cut-off frequency.
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