Type II Cepheids Pulsating in the First Overtone from the OGLE Survey

TL;DR

This paper reports the discovery of the first known type II Cepheids pulsating solely in the first overtone, identified in the OGLE survey, and discusses their properties, rarity, and modeling challenges.

Contribution

It presents the first identification of single-mode first-overtone BL Her stars and provides non-linear models explaining their light variations and mass-luminosity discrepancies.

Findings

Two first-overtone BL Her stars discovered in OGLE data

Models suggest these stars are more massive than typical BL Her stars

Higher mass models conflict with observed luminosities

Abstract

We report the discovery of the first type II Cepheids (BL Herculis stars) pulsating solely in the first overtone. We found two such objects among tens of millions of stars regularly observed by the OGLE survey in the Large Magellanic Cloud. Our classification and the pulsation mode identification is based on the position of these stars on the period-luminosity and color-magnitude diagrams and on the light curve analysis. We discuss why single-mode first-overtone BL Her pulsators must be very rare. For the two discovered stars we present non-linear models that successfully reproduce their light variation. These models indicate that both first-overtone pulsators should be more massive than it is typically assumed for BL Her stars, i.e. their masses should be above 0.75 Solar masses. However, the higher mass requires higher luminosity to match the observed periods of the stars, which is inconsistent with observations.

Tables

Table 1. Table 1: First-overtone BL Her stars in the LMC

Identifier	R.A.	Dec.	Period	$⟨I⟩$	$⟨V⟩$	$A\left(I\right)$
	[J2000.0]	[J2000.0]	[d]	[mag]	[mag]	[mag]
OGLE-LMC-T2CEP-290	04:51:11.61	$-$69:00:33.0	0.9321202	17.869	18.520	0.238
OGLE-LMC-T2CEP-291	05:42:38.85	$-$61:59:44.3	0.8175507	18.135	18.647	0.417

Table 2. Table 2: Physical parameters of non-linear models computed for OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 that converged to full amplitude single-periodic first-overtone pulsation.

ID	mass [ M⊙]	$\left[\mathrm{Fe}/H\right]$	$T_{\mathrm{eff}}$ [K]
OGLE-LMC-T2CEP-290	0.85	$-0.5$	6450–6500
		$-1.0$	6350–6500
		$-1.5$	6400–6450
	0.90	$-0.5$	6350–6550
		$-1.0$	6300–6500
OGLE-LMC-T2CEP-291	0.75	$-0.5$	6450–6550
		$-1.0$	6400–6550
	0.80	$-0.5$	6350–6600
		$-1.0$	6350–6550
	0.85	$+0.0$	6400–6550
		$-0.5$	6350–6600
		$-1.0$	6400–6500
	0.90	$+0.0$	6350–6600
		$-0.5$	6350–6550

Full text

stars: variables: Cepheids — stars: oscillations — stars: Population II — Magellanic Clouds

††journal: ApJ

1 Introduction

The standard model of stellar pulsations (Eddington, 1926) predicted that stars may pulsate not only in the fundamental mode but also in the overtone modes. Edgar (1933) was the first who theoretically studied this idea in detail, while Schwarzschild (1940) first suggested that RR Lyrae stars of type c (Bailey, 1902) are the first-overtone pulsators. The physical nature of the so-called sinusoidal or s-Cepheids has been a matter of debate for years (e.g. Arp, 1960; Ivanov & Nikolov, 1976; Connolly, 1980; Antonello et al., 1990; Gieren et al., 1990) until large-scale microlensing surveys: EROS (Beaulieu et al., 1995), MACHO (Alcock et al., 1995) and OGLE (Udalski et al., 1999) presented large samples of classical Cepheids in the Magellanic Clouds and unambiguously proved that s-Cepheids are the first-overtone pulsators. Modern catalogs of classical Cepheids contain single- and multi-mode variables with the first, second and even the third overtone excited (e.g. Soszyński et al., 2015).

In the virtually complete collection of Cepheids and RR Lyr stars (collectively referred to as classical pulsators) in the Magellanic Clouds published by the OGLE survey (Soszyński et al., 2015, 2017a), the overtone pulsators constitute a significant fraction of the total population. About 28% of all RR Lyr stars, 31% of anomalous Cepheids, and 46% of classical Cepheids have the overtone modes excited (including multi-mode pulsators). Type II Cepheids are an exception in the family of classical pulsators – until recently all known variables of this class were pure fundamental-mode pulsators.

The first two double-mode type II Cepheids pulsating simultaneously in the fundamental and first-overtone modes were discovered by Smolec et al. (2018), who identified them among nearly one thousand type II Cepheids detected by the OGLE survey in the Galactic bulge (Soszyński et al., 2017b). These two unique objects have (fundamental-mode) periods equal to 1.04 d and 1.18 d, so they belong to the shortest-period subclass of type II Cepheids – BL Herculis stars – just above the conventional borderline (1 d) between RR Lyr stars and type II Cepheids. The ratios of the first-overtone to the fundamental-mode periods for these two stars are almost identical and are equal to about 0.705. Recently, Udalski et al. (2018) reported the discovery of two other double-mode BL Her stars in the Galactic bulge and disk.

Inspired by these discoveries, we decided to search the OGLE photometric databases for single-mode first-overtone BL Her stars. The best environment to perform such a search is the Magellanic System, in particular the Large Magellanic Cloud (LMC), because it contains rich populations of spatially resolvable stars at approximately the same distance from us. This allow us to precisely locate pulsating stars on the period–luminosity (PL) diagram to find out in which mode they pulsate. The long-term OGLE photometry is useful for studying the light curve morphology which is also an invaluable discriminator between fundamental and overtone pulsators.

2 Observational Data and Candidate Selection

The OGLE database contains light curves of over 75 million stars in the Magellanic Clouds. These data have been collected with the 1.3-m Warsaw Telescope located at Las Campanas Observatory in Chile. The observatory is operated by the Carnegie Institution for Science. Since 2010, the Warsaw Telescope has been equipped with the 256 Megapixel mosaic CCD camera with a field of view of 1.4 square degrees at a scale of 0.26″/pixel. OGLE uses the V- and I-band filters from the Johnson-Cousins photometric system, but most of the observations (typically 700 points per star) have been carried out with the I passband. Details of the OGLE instrumental setup, data reduction and calibration can be found in Udalski et al. (2015).

Candidates for the first-overtone BL Her stars have been sought among sources observed since 2010 by the OGLE-IV survey. First, we used the Fnpeaks code111http://helas.astro.uni.wroc.pl/deliverables.php?active=fnpeaks to compute the Fourier amplitude spectra for all I-band light curves obtained by OGLE in the Magellanic System to determine periods with corresponding amplitudes and signal-to-noise ratios.

Second, we defined PL relations obeyed by hypothetical first-overtone type II Cepheids, separately for the LMC and Small Magellanic Cloud (SMC). We used the $\log{P}$ vs. $W_{I}$ and $\log{P}$ vs. I-band luminosity planes, where $W_{I}$ is the reddening-independent Wesenheit index, defined as $W_{I}=I-1.55(V-I)$, and $I$ and $V$ are the intensity-averaged mean magnitudes in these filters. In Figure 1, we present the period-Wesenheit (PW) and PL diagrams for classical pulsators in the LMC (Soszyński et al., 2017a, 2018): classical Cepheids (blue points), anomalous Cepheids (green points), type II Cepheids (pink points), and RR Lyr stars (orange points). Lighter and darker colors indicate fundamental-mode and overtone-mode pulsators, respectively. The dashed red lines show the expected PW and PL relations for the first-overtone BL Her stars. These lines have the same slopes as the relations for the fundamental-mode type II Cepheids (plotted with the solid red lines) and are shifted toward the shorter periods, corresponding to the period ratio of 0.7 between both modes.

Third, we visually inspected all light curves in both Magellanic Clouds meeting the following criteria: (i) the primary period in the range 0.7–3 d (corresponding to 0.7 of the period range of fundamental-mode BL Her stars 1–4 d), (ii) I-band luminosity and $W_{I}$ Wesenheit index that placed the star up to 0.3 mag above or below the red dashed lines shown in Figure 1 (which is comparable to maximum dispersions of the PL relations obeyed by other types of classical pulsators), (iii) I-band peak-to-peak amplitude larger than 0.15 mag (to avoid low-amplitude non-radial pulsators and other types of variable stars). Using these criteria we selected several thousand variable stars in both Magellanic Clouds and carefully examined their light curves.

As a result, we chose two pulsating variables which we believe are the first-overtone BL Her stars. Both objects are located in the LMC. Their positions on the PW and PL diagrams are shown by red circles in Figure 1. In Figure 2, we present their I-band light curves phased with periods 0.932120 d and 0.817551 d, respectively. No additional periodicities have been found in these light curves.

These two stars have been added to the OGLE Collection of Type II Cepheids in the LMC (Soszyński et al., 2018). Their time-series OGLE-IV photometry and finding charts are available through the FTP site:

ftp://ftp.astrouw.edu.pl/ogle/ogle4/OCVS/lmc/t2cep/

Both objects received unique identifiers which follow the scheme proposed in the OGLE Collection: OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291. Table 1 contains their basic parameters: identifiers, J2000.0 equatorial coordinates, pulsation periods, mean magnitudes in the I and V bands, and I-band amplitudes.

3 Discussion

3.1 Shapes of the Light Curves

In this section, we will provide evidence that OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are indeed stars pulsating in the first-overtone. First, based on the light curve morphology (Figure 2), there is no doubt that both objects are radially pulsating stars. Theoretically, spotted rotating stars may produce light curves that mimic any type of periodic variables, including pulsators, but such light curves are unstable and change shapes on a time scale of months as the spot coverage changes on the stellar surface. Meanwhile, the light curves of our objects have been stable for years: at least from 2001 for OGLE-LMC-T2CEP-290 (which has been monitored by the OGLE-III and OGLE-IV surveys) and from 2010 for OGLE-LMC-T2CEP-291 (observed during the OGLE-IV project). Characteristic shapes of the light curves indicate that both stars are pulsating variables.

Second, the position of our objects on the PW and PL diagrams (Figure 1) is close to the expected relation for the overtone BL Her stars. Although OGLE-LMC-T2CEP-290 is slightly brighter, while OGLE-LMC-T2CEP-291 slightly fainter than the theoretical relation for the overtone type II Cepheids, but their dispersion around this relation is typical for other classical pulsators in the LMC.

As one can see in Figure 1, the PW and PL relations for the first-overtone BL Her stars partly overlap the relations for the fundamental-mode RR Lyr variables (RRab stars) and fundamental-mode anomalous Cepheids. For this reason, both our candidates for overtone pulsators have already been included in the OGLE Collection of Variable Stars and classified as a fundamental-mode anomalous Cepheid (OGLE-LMC-T2CEP-290) and as an RRab star (OGLE-LMC-T2CEP-291). However, a closer look at the light curves of our objects indicates that they are not fundamental-mode pulsators.

In Figure 3, we present three light curves of typical fundamental-mode anomalous Cepheids and three light curves of RRab stars from the LMC. Here, we selected variables with periods similar to the periods of our candidates for the first-overtone BL Her stars (around 0.82 or 0.93 days). The difference between the light curves in Figures 2 and 3 is visible to the naked eye. In particular, we draw the reader’s attention to the shape of the light curves close to the minimum brightness. Anomalous Cepheids with these periods exhibit a small bump just before the sharp minimum. RRab stars also have sharp minima, while both our candidates for the first-overtone BL Her stars show round minima.

Figure 4 displays a sample of I-band light curves belonging to the first-overtone classical pulsators in the LMC: classical Cepheids, anomalous Cepheids and RR Lyr stars (RRc variables). We selected typical overtone variables covering a wide range of pulsation periods. The morphology of the light curves varies from star to star, but one feature is common for all stars with the overtone modes excited – they have round minima of their light curves. This is a very strong argument that OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 belong to the same (first-overtone) class of pulsating stars.

The coefficients derived from the Fourier decomposition of light curves provide a quantitative description of their morphology. We fitted the cosine Fourier series to the I-band light curves of our variables and derived the $\phi_{21}$, $\phi_{31}$, $R_{21}$, and $R_{31}$ coefficients (Simon & Lee, 1981). Figure 5 compares these parameters for first-overtone classical pulsators in the LMC. The light curves of our candidates seem to be the most similar to the first-overtone classical Cepheids, although at the considered period range, Fourier parameters for anomalous Cepheids and classical Cepheids partially overlap. This similarity confirms that OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are the first overtone pulsating stars, although they do not fit perfectly in each diagram which is expected since they are members of a different class of pulsators than classical and anomalous Cepheids. Our candidates have smaller Fourier phases $\phi_{21}$ and $\phi_{31}$ (upper panels of Figure 5) than typical anomalous Cepheids with the same periods and slightly larger amplitude ratios $R_{21}$ and $R_{31}$ (lower panels of Figure 5) than most of classical Cepheids. The Fourier parameter diagrams nicely confirm the first-overtone pulsation of OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 and, together with the PL and PW diagrams, discriminate between variability classes.

3.2 Position in the Color–Magnitude Diagram

Another indicator of the pulsation mode is the position of a star in the color–magnitude diagram. It is known that the overtone pulsators occupy the blue side of the instability strip. Their $(V-I)$ colors are on average 0.10–0.15 mag smaller than for the fundamental-mode Cepheids and RR Lyr stars, although there is an overlap between different pulsation modes. Additionally, the apparent color indices of individual objects may be strongly affected by the interstellar reddening or blending by an unresolved companion, which limits the applicability of the color criterion for determining the pulsation modes.

In the upper panel of Figure 6, we present the $(V-I)$ vs. $I$ color–magnitude diagram for classical pulsators in the LMC. As in Figure 1, darker colors indicate overtone variables. OGLE-LMC-T2CEP-291 has an apparent color index $(V-I)=0.51$ mag, which agrees with typical colors of other overtone classical pulsators and is bluer than most of the fundamental-mode BL Her stars. This is another direct proof that OGLE-LMC-T2CEP-291 pulsates in the first-overtone mode.

However, the color index of OGLE-LMC-T2CEP-290 (0.65 mag) places this star closer to the fundamental-mode pulsators. This fact can be explained by a high interstellar reddening toward this object. Lower panels of Figure 6 present color–magnitude diagrams for all stars (included in the OGLE-IV database) within a radius of 10 arcmin (for OGLE-LMC-T2CEP-291) and 5 arcmin (for OGLE-LMC-T2CEP-290) around both candidates for the overtone type II Cepheids. The surroundings of OGLE-LMC-T2CEP-291 (lower left panel of Figure 6) are characterized by well-defined red clump with the mean apparent color index of $(V-I)=0.97$ mag. On the other hand, in the vicinity of OGLE-LMC-T2CEP-290 (lower right panel), the red clump is significantly elongated toward redder and fainter stars, which indicates high and differential reddening toward these regions. The mean apparent $(V-I)$ color index of the red clump stars in this area cannot be precisely determined – we can only specify that it is included in the range from 1.01 to 1.22 mag. It translates into a range from 0.04 to 0.25 mag of higher reddening toward OGLE-LMC-T2CEP-290 than to OGLE-LMC-T2CEP-291, which fully explains the redder color of the former object.

3.3 Spatial Location

Figure 7 displays the sky map of the LMC with classical pulsators marked with the same symbols as in Figure 1. OGLE-LMC-T2CEP-290 is located closer to the LMC center, on the western edge of the LMC bar, while OGLE-LMC-T2CEP-291 is located in the northern outskirts of the LMC, where classical Cepheids are absent, but the old stellar population (RR Lyr stars) is still common. This confirms that at least OGLE-LMC-T2CEP-291 belongs to the old population.

What is the distance of our candidates for the first-overtone BL Her stars? So far we have assumed that both objects are located in the LMC and their position on the apparent PL diagram (Figure 1) reflects their position in the absolute PL plane. However, there is a possibility that we deal with first-overtone classical or anomalous Cepheids located behind the LMC. Taking into account the position of both stars in the extinction-free PW diagram, we may estimate that if they are the first-overtone classical Cepheids, they should be located about twice as far as the center of the LMC. It seems very unlikely that stars belonging to the young stellar population (classical Cepheids) exist in the intergalactic space. In turn, assuming that OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are the first-overtone anomalous Cepheids, they should be located 50-60% farther than the LMC. Taking into account that the light curves of our objects (quantified by the Fourier coefficients; Figure 5) are different than light curves of typical first-overtone anomalous Cepheids with the same periods, such a possibility also seems unlikely.

3.4 First-Overtone BL Her Stars from Theoretical Perspective

While single-periodic first-overtone pulsation is common among classical Cepheids and RR Lyr stars, it has not been detected so far in any type II Cepheid. Here we explain why first-overtone pulsation for type II Cepheids must be very rare. In Figure 8, we show the pulsation and evolutionary scenarios for classical Cepheids (left panel) and for RR Lyr stars and the shortest-period type II Cepheids – BL Her stars (right panel). Instability strips for the fundamental mode (F mode; solid blue and solid red lines) and for the first-overtone mode (1O mode; gray-shaded area) were computed using linear convective codes of Smolec & Moskalik (2008a). In all computations presented in this paper parameters of the turbulent convection model are the same as we used in the successful modeling of a period-doubled BL Her star (Smolec et al., 2012). For classical Cepheids, the model grid was constructed assuming mass–luminosity relation inferred from evolutionary models of Anderson et al. (2016) for ${\rm[Fe/H]}=-0.5$. In the right-hand panel (RR Lyr & BL Her stars) we assumed a constant mass of $0.60{\,}{\rm M}_{\odot}$ and ${\rm[Fe/H]}=-1.0$. Evolutionary tracks for classical Cepheids are from Bertelli et al. (2009) ($Z=0.008$) and evolutionary tracks for horizontal-branch stars are from Dotter et al. (2008) (${\rm[Fe/H]}=-1.0$).

We note that model calculations are sensitive e.g. to the physical parameters, treatment of the convection, or details of the evolutionary modeling. For example, different extent of overshooting during main sequence evolution can shift the evolutionary tracks for Cepheids in luminosity and affect the extent of the blue loops. By adopting different parameters for turbulent convection model in pulsation calculations, the location of the instability strips may, e.g. slightly shift in effective temperature or in luminosity, but the overall shape and features of the instability strip, like limited extent of 1O instability strip in luminosity, will be the same. Here we just discuss the qualitative pulsation and evolutionary picture and these details are inessential for our discussion.

For classical Cepheids (left panel of Figure 8) we can identify a domain in which only 1O is linearly unstable (dark-gray shaded area). If a star moves toward lower effective temperatures (first and third crossing of the instability strip) it first enters the domain in which only 1O is linearly unstable and single-periodic 1O pulsation is the only possibility. Then, it enters the domain in which both modes, F and 1O, are linearly unstable (light-gray shaded area), and finally a domain in which F mode pulsation is the only possibility. What happens in the domain in which both modes are linearly unstable is dictated by non-linear mode selection (see, e.g. Smolec & Moskalik, 2008b). The star that enters this domain from the hot side may continue the single-periodic 1O pulsation until it switches to F mode single-periodic pulsation somewhere within the domain. Similarly, the star that enters this domain from the cool side may continue the F mode pulsation, until it switches to 1O single-mode pulsation somewhere within the domain. In a part of the HR diagram within light-gray shaded area the so-called either-or scenario is possible: the star may pulsate either in 1O mode or in F mode depending on the direction of evolution. The other possible scenario is double-mode pulsation: simultaneous pulsation in both F and 1O modes. Only with non-linear calculations this mode selection problem can be addressed. It is, however, clear that over significant part of the HR diagram single-periodic 1O pulsation in classical Cepheids is possible. First-overtone classical Cepheids, as well as double-mode F+1O classical Cepheids are common.

For RR Lyr stars (right panel of Figure 8; stars with fundamental-mode periods below $1$ d, i.e. below the thick long-dash black line) the scenario is qualitatively the same as just outlined for classical Cepheids. We can identify the domain in which 1O pulsation is the only possibility (hottest pulsators; dark-gray shaded area) and a domain in which both 1O and F modes are linearly unstable (light-gray shaded area), in which either single-periodic 1O, or single-periodic F mode pulsation, as well as double mode F+1O pulsation are possible. Indeed, both single-periodic 1O pulsators (RRc stars) and double-mode pulsators (RRd stars) are common (as well as single-periodic F mode pulsators, RRab stars).

At this point it is essential to point that there is a maximum period for 1O RR Lyr pulsators. Considering the most numerous OGLE collection of RR Lyr variables (Soszyński et al., 2014, 2016) the longest possible 1O periods are approximately $0.539$, $0.495$ and $0.487$ d for RRc stars of the Galactic bulge, LMC, and SMC, respectively and $0.451$, $0.521$ and $0.524$ d for RRd stars of the same stellar systems. These numbers are all quite similar; differences often rely on a single record-holder star. In 99 % of the above considered stars from the OGLE collection, 1O period is shorter than $0.44$ d – a value marked with thick short-dashed line in Figure 8. Observations clearly indicate that above this line nearly all RR Lyr stars pulsate in F mode only, despite 1O can still be linearly unstable there. We note that above the discussed line, 1O domain is essentially engulfed by the domain in which F mode is unstable. Consequently, before entering the domain in which both F and 1O modes are linearly unstable, a star was necessarily single-periodic F mode pulsator.

The lack of long-period RRc/RRd stars indicates that non-linear mode selection favors the F-mode-only pulsation also in the domain in which both modes are linearly unstable. A star that was pulsating in the F mode continues the F mode pulsation also after entering the domain in which 1O mode becomes linearly unstable. RR Lyr stars and BL Her variables are close siblings (e.g. Iwanek et al., 2018). BL Her stars are just slightly less massive, start the HB evolution at higher effective temperatures and cross the instability strip at larger luminosities. The instability strip displayed in Figure 8 was computed for $M=0.6{\,}{\rm M}_{\odot}$, however for lower masses, the 1O instability strip shrinks and extends to lower luminosities (see Figure 9, bottom panels) so long-period single-periodic 1O pulsation is even more excluded for BL Her stars.

Still, as we determined on observational basis, OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 with much longer periods are single-periodic 1O pulsators. To understand the nature of these two stars we have computed a small model survey covering a much larger range of masses than is typically considered for BL Her-type stars. Masses in the grid cover $0.5-0.9{\,}{\rm M}_{\odot}$ range with $0.05{\,}{\rm M}_{\odot}$ step. Their metallicities are in the $-2.5,\ldots,+0.0$ range with 0.5 dex step.

Instability strips for selected models are plotted in the HR diagrams in Figure 9, along with lines of constant period equal to the periods of OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291. In addition, thick long-dashed line is placed at constant fundamental-mode period equal to 1 d and thus separates the RR Lyr and BL Her classes. We find that the lower the mass the shorter the extension of 1O instability strip in luminosity. In particular, for $M=0.5{\,}{\rm M}_{\odot}$, there are no models for OGLE-LMC-T2CEP-290 in which the first overtone is linearly unstable. The line of constant period corresponding to pulsation period of OGLE-LMC-T2CEP-291 crosses the 1O instability domain at its tip. For larger masses, 1O is linearly unstable for both stars over a significant range of effective temperatures. We also note that along the discussed lines of constant 1O period, 1O instability domains are always engulfed by F mode instability domain. Systematics with metallicity is difficult to spot in Figure 9. We note that with increasing metallicity the instability strips shift toward lower effective temperatures. Also, the larger the metallicity the narrower the 1O instability strip. Based on linear models we may expect that the larger the mass the higher the chances for 1O pulsation for OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291. However, only with non-linear calculations one may study for which physical parameters stable large amplitude 1O pulsation is possible.

Non-linear models were computed using the codes of Smolec & Moskalik (2008a). Models were computed for the same grid of masses and metallicities as in the linear model survey discussed above. For a given mass and metallicity, the models were computed along the lines of constant 1O periods, corresponding to the periods of OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291, with a $50$ K-step in effective temperature. These models were computed only across the domain in which the first-overtone is linearly unstable. As it is clear from Figure 9, the fundamental mode is necessarily unstable as well then. The static models were perturbed with the first-overtone scaled velocity eigenfunction calculated for each model during linear analysis, and time integration was conducted for at least $6\,000$ pulsation cycles.

During the model integration, after initial transient phase in which first-overtone pulsation was present, the majority of models switched into single-mode fundamental-mode pulsation. It was the case for all models with masses below $0.75{\,}{\rm M}_{\odot}$. It well agrees with the considerations presented above: with parameters expected for typical BL Her stars single-mode first-overtone pulsation is not possible. As mass is increased in the models we observe that the switching from the first-overtone pulsation to fundamental-mode pulsation occurs later during the integration. For models of OGLE-LMC-T2CEP-291 (with shorter pulsation period, $P\approx 0.818$ d), single-mode first-overtone pulsation becomes possible for $M\geq 0.75{\,}{\rm M}_{\odot}$. For slightly more luminous and longer-period models of OGLE-LMC-T2CEP-290 ($P\approx 0.932$ d) first-overtone pulsation becomes possible only for $M\geq 0.85{\,}{\rm M}_{\odot}$. In Table 2 we provide the physical parameters of the models that converged for single-mode 1O pulsation. We find that 1O pulsation is not possible for the lowest considered metallicities, i.e. for $-2.5$ and $-2.0$. For ${\rm[Fe/H]}=-1.5$ only a single model converged to 1O pulsation ($M=0.85{\,}{\rm M}_{\odot}$). 1O pulsation is possible for ${\rm[Fe/H]}\geq-1.0$. For the highest considered mass 1O models are present also for solar metallicity. The higher the mass, the wider the domain of 1O pulsation.

A rather large domains over which models converged to full amplitude 1O pulsation may indicate that such pulsation should be commonly observed. We recall however, that the way we initialized model integration – low-amplitude perturbation with scaled velocity eigenvector of 1O mode – favors 1O pulsation. As it is clear from Figure 9, the star that enters the domain in which 1O mode is linearly unstable is already a large amplitude F mode pulsator and may continue the F mode pulsation in the domain in which both F and 1O modes are unstable. In principle, using non-linear calculation one can study the stability of full-amplitude F mode pulsation with respect to perturbation in 1O mode. This is, however, very time-consuming as it requires several integrations of the same model with different initial conditions (see, e.g. Smolec & Moskalik, 2008b). Such analysis is beyond the scope of the present paper, but it is planned.

The computed bolometric light curves were transformed to the $I$-band via static Castelli & Kurucz (2004) model atmospheres. In Figure 2, we compare the observed light curves with the best matching model, i.e. the model in which relative differences of low-order Fourier coefficients between the model and observed light curve are the lowest. Physical parameters of these models are given in the panels. We note that the pulsation amplitude in the best model for OGLE-LMC-T2CEP-291 is nearly half of that observed. We comment on this difference more in the following. We note that the light curves of all computed first-overtone-models are all very similar to each other and that plotted in Figure 2 may be regarded as representative. The match between the model and observed light curves is reasonable. In the models, the change of inclination on the ascending branch, after the minimum brightness is quite pronounced, while in the observed light curves it is well visible only for OGLE-LMC-T2CEP-290. To compare the models and light curves in a more quantitative way, in Figure 5 we have also plotted the Fourier parameters of the best-matching models (large crosses). The match for OGLE-LMC-T2CEP-290 is remarkable. For OGLE-LMC-T2CEP-291 the match for Fourier phases is very good, while amplitude ratios are a bit lower, which is expected, as pulsation amplitude is also lower (Figure 2).

Although the light variation of OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 can be reproduced reasonably well, the first-overtone non-linear models we have just discussed are inconsistent with observations regarding their luminosity. When the mass is increased, the luminosity must also be increased to match the periods of the two modeled stars (see Figure 9). Consequently, our models are both more massive and more luminous than expected for BL Her stars. In the PW plane (Figure 1), the models are placed at a position characteristic for fundamental-mode anomalous Cepheids, in between position typical for first-overtone anomalous Cepheids and position expected for first-overtone BL Her stars. The best-matching models for OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are by 0.36 and 0.58 mag more luminous than observed, respectively.

Our initial model survey indicates that indeed the light curve shape observed for OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 is characteristic for single-mode first-overtone pulsation. However, single-mode first-overtone pulsation becomes possible only when mass is significantly increased above the values expected for BL Her stars. It may indicate that our stars are a product of specific mass transfer event during binary evolution. Binarity is often invoked as plausible explanation for the properties of peculiar W Vir stars or of anomalous Cepheids (e.g. Soszyński et al., 2017b). In this scenario, however, the stars should be more luminous than observed. Pulsation modeling of OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 remains a puzzle at the moment.

We stress that our models survey is very limited; a thorough analysis that may help to resolve the raised issues, is planned. In particular, the following should be addressed in a more detailed model survey. (i) Full mode selection analysis should be done. It requires integration of each model with various initial conditions and analysis of the resulting hydrodynamic trajectories with amplitude equation formalism. (ii) Other sets of convective parameters should be investigated. In particular, models with decreased eddy-viscous dissipation. Such models should have larger pulsation amplitudes and hence should match the observations of OGLE-LMC-T2CEP-291 better. More importantly, the extent of the instability domains and mode selection strongly depend on eddy-viscous dissipation and other parameters that enter the convection model used in the code. By varying these parameters, we may hope for first-overtone pulsation at lower masses and hence at lower luminosities. (iii) Extension of the model survey to even higher masses, up to that considered for anomalous Cepheids ($\sim 1.5{\,}{\rm M}_{\odot}$), is desired.

4 Conclusions

We reported here the discovery of two candidates for single-mode first-overtone type II Cepheids. OGLE-LMC-T2CEP-291 meets all observational criteria that can be checked using the OGLE data: positions in the PL and PW diagrams, light curve morphology, and $(V-I)$ color index. OGLE-LMC-T2CEP-290 meets the first two conditions, but it is too red as for the first-overtone pulsator, however this fact can be explained by a higher interstellar reddening toward this star. We believe that our detection is very reliable, since the light curves of our two candidates have definitely different shapes than fundamental-mode pulsators located in this region of the PL diagram. Spectroscopic follow-up observations could bring new arguments for or against our classification.

To our knowledge, OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are the first known type II Cepheids pulsating solely in the first-overtone. McCollum et al. (1997) suggested that a bright BL Her star VY Pyx is an overtone pulsator, however a glance at the light curve of this objects (Sanwal & Sarma, 1991) reveals sharp minima, which is a characteristic feature of the fundamental-mode pulsators. Also the Hipparcos parallax place VY Pyx on the PL relation obeyed by the fundamental-mode type II Cepheids (Feast et al., 2008).

With the help of pulsation models we explained why first-overtone pulsation of BL Her stars must be very rare, or is hardly possible, at least for masses typically assumed for these stars. For masses $\sim 0.5-0.6{\,}{\rm M}_{\odot}$ the linear instability strip for the first-overtone mode does not extend to sufficiently high luminosities to secure first-overtone periods well above $0.7$ d, in the range expected for BL Her variables. As mass is increased ($\sim 0.6-0.7{\,}{\rm M}_{\odot}$), this restriction is no longer valid: linear first-overtone instability strip extends to sufficiently high luminosities, at least to warrant pulsation periods observed in OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291. However, the first-overtone instability strip is entirely engulfed within fundamental-mode instability strip then. As these are the masses expected also for RR Lyr stars, observations, namely the first-overtone periods observed in RRc and RRd stars (always below $\approx 0.5$ d), point that non-linear mode selection favors fundamental-mode pulsation in the high luminosity part of the joint instability strip of the two radial modes. This was fully confirmed with the help of non-linear pulsation calculations. The model integration started with first-overtone perturbation always led to fundamental-mode finite amplitude pulsation. Only for $M\geq 0.75{\,}{\rm M}_{\odot}$ the model integration finally ended in stable first-overtone pulsation. We stress that the best matching model for OGLE-LMC-T2CEP-290 ($0.9{\,}{\rm M}_{\odot}$) matches the observed light curve remarkably well. The best matching model for OGLE-LMC-T2CEP-291 ($0.8{\,}{\rm M}_{\odot}$) also well reproduces the observed light variation, only the amplitude is about twice as low as observed. The successful modeling of the light variation provides yet another strong argument that OGLE-LMC-T2CEP-290 and OGLE-LMC-T2CEP-291 are first-overtone pulsators.

The increased mass has its consequence, however. Evolution of single stars does not lead to such massive horizontal branch stars. Although evolution in a binary system would provide a solution, higher mass implies higher luminosity, which is inconsistent with observations. More extensive modeling is planned to resolve this puzzle.

This work has been supported by the National Science Centre, Poland, grant MAESTRO no. 2016/22/A/ST9/00009. The OGLE project has received funding from the Polish National Science Centre grant MAESTRO no. 2014/14/A/ST9/00121. IS and RS are supported by the Polish National Science Centre grant OPUS no. DEC-2015/17/B/ST9/03421.