Tides Cepheid
Elnora Heidrick
Classical Cepheids are intermediate-mass, evolved stars which have been observed for centuries ever since Cepheid variability was discovered by John Goodricke in 1784 (Goodricke & Bayer 1786). The source of the variability was a long-standing mystery, one suggestion was that Cepheids are eclipsing binary stars. However, the first distance measurements to nearby Cepheids implied that a Cepheid light curve could only be consistent with an eclipsing binary if the orbital separation was less than the stellar radius, negating the binary hypothesis. While disproved, the binary hypothesis persisted until the 1920s (Jeans 1925), even though Eddington (1917) demonstrated that Cepheid variability is due to radial pulsation.
It is radial pulsation that makes Cepheid variable stars important for stellar physics, extragalactic astronomy and cosmology. Leavitt (1908) discovered the Cepheid period-luminosity relation (Leavitt Law) from observations of Large Magellanic Cloud (LMC) Cepheids, which was applied to extragalactic observations to measure the Hubble constant (Lematre 1927; Hubble 1929), thus founding modern cosmology.
Measurements of Cepheid properties such as mass, radius and luminosity constrain the detailed physics of stellar evolution and pulsation. For instance, theories of helium fusion (Morel et al. 2010) and mass loss (Neilson et al. 2012a,b) have been constrained by measurements of period change, while mass measurements constrain theories of convective core overshooting (Keller & Wood 2006; Neilson et al. 2011). These results have implications for late-stage stellar evolution and supernovae, as well as constraining the main sequence evolution of their progenitors.
Observations are providing insight into the orbital period and eccentricity distributions of binary Cepheids in the Galaxy. Evans et al. (2005) presented a list of single-line spectroscopic binaries with measured orbital periods and eccentricities, as well as updated orbital parameters for the Cepheid Polaris (Evans et al. 2008), based on a literature review of all known Galactic Cepheids. That binary sample is complete with the exception of small mass ratios (< 0.2) and long-period orbits (>10 yr) and is the most complete sample of Galactic Cepheid binarity. Szabados & Nehz (2012) argued that the Galactic Cepheids X Pup and XX Sge are also spectroscopic binaries based on 75 yr of observations. Szabados et al. (2013b,a) analyzed spectroscopic observations to detect binary companions for a sample of Galactic Cepheids in the southern hemisphere. Li Causi et al. (2013) observed the companion of the Cepheid X Sgr using interferometric observations. Similarly, Gallenne et al. (2013, 2014) discovered, using interferometric observations, that the Cepheids V1334 Cyg and AX Cir also have companions. Evans et al. (2005) noted that about 35% of Galactic Cepheids have at least one spectroscopic companion, and about 44% of those have more than one companion. There are no known Galactic Cepheids with a companion with orbital period less than one year (Evans et al. 2011).
Observational studies of Cepheid duplicity in the LMC are also beginning to yield results. Three eclipsing binaries were discovered in the OGLE-III survey of LMC Cepheids (Soszynski et al. 2008) while Szabados et al. (2012) noted there are four known spectroscopic Cepheid binaries. Szabados et al. (2012) also attempted to use velocity and B- and V-band amplitudes to search for new binary Cepheids. From their sample of 43 Cepheids, Szabados et al. (2012) detected seven additional binary Cepheids. They suggested that the binary fraction of Cepheids in the LMC may be less than that of the Milky Way, but the result still suffers from significant observational bias because of their method.
The purpose of this work is to explore Cepheid duplicity from the perspective of theoretical binary stellar evolution models and to compare predicted Cepheid binary properties with those observed in the Galaxy and LMC. As such we compute the probable minimum orbital period for Galactic Cepheids from the modeled orbital period distribution. We also predict the number of Cepheids in eclipsing binary systems that would be detectable in the OGLE-III survey to compare with the observed number. This work is a first step in understanding the role of binarity in Cepheid stellar evolution. We explore the distribution of binary systems containing a Classical Cepheid as the primary component from population synthesis calculations. We discuss the synthesis code and describe our assumptions in Sect. 2. In Sect. 3, we present our results for Galactic and LMC metallicities. We compare these results to the observed binary fraction in Sect. 4 and discuss the role of tidal evolution and mass transfer on the evolution of stars from the red giant branch to the blue loop. We conclude in Sect. 5.
We label a star as a Cepheid when one of the stars evolves onto the Cepheid instability strip as determined by Bono et al. (2000b). This includes core helium-burning stars, and stars evolving across the Hertzsprung gap. However the latter population is expected to be small because their evolutionary timescale is much shorter than for later crossings of the instability strip. These first crossing Cepheids contribute less than a few percent to the total Cepheid population. We also count binary systems in which the secondary star is a Cepheid. If a Cepheid is the secondary then the primary is initially more massive, hence is more likely to have evolved into a white dwarf star or exploded as a supernova. Such a Cepheid binary system will be rare and difficult to observe because the relic companion will be much dimmer than the Cepheid. However, in rare cases the initially more massive primary star will accrete onto the secondary during main sequence evolution, and the initial primary may not evolve to be a white dwarf or neutron star before the other star becomes a Cepheid. These systems are also counted, but not considered in Sect. 3.
Fig. 1Orbital period distribution of stars residing inside the Cepheid instability strip at solar metallicity, Z = 0.02, according to our population synthesis models. (Left) Binary systems for the entire period range, systems with an orbital period longer than 11 yr are not plotted. (Right) Systems with orbital periods shorter than 1 yr. The dotted histogram represents all systems evolving across the instability strip while the solid histogram represents those systems identified as Cepheids, i.e., with no significant mass transfer from the primary to secondary.
For further comparison, we plot the probability map of the orbital period and eccentricity for Galactic Cepheids in Fig. 2, along with the periods and eccentricities for a sample of Galactic Cepheid binaries from Evans et al. (2005). Many of these spectroscopic binaries were detected from ultraviolet observations. All Cepheid binaries in the sample are spectroscopic binaries with the exception of Polaris which is an astrometric binary (Kamper 1996; Evans et al. 2008). One-half of the Cepheids in the sample have orbital periods between one and two years and most have eccentricities less than 0.5. There may be an observational bias towards short-orbital period Cepheids because they are more easily identified as spectroscopic binaries.
There is some agreement between our model and the distribution of Galactic Cepheid binaries. The observed number ratio between binary Cepheids with orbital periods < 2 yr and those 2 2 yr. Observed binaries with Porb> 2 yr appear to have more circular orbits than our model predicts. Specifically, there appear to be five systems with orbital periods longer than two years and small eccentricity (< 0.25), otherwise the agreement between our model and observations is remarkable. The difference for those five systems is likely because of the assumed initial eccentricity distribution and resolution of the models but, without a greater number of observed binary Cepheids to which we can compare, it is difficult to quantify the eccentricity discrepancy between the predictions and observations and whether it is significant.
Fig. 2Comparison of the orbital period and eccentricity distributions from our population synthesis models and properties of observed Galactic Cepheid binary systems (Evans et al. 2005), denoted with dots. The colour scale represents the total probability that a binary of given properties exists assuming constant star formation.
We also compare population synthesis models assuming metallicity Z = 0.008 with the OGLE-III survey of classical Cepheids and plot the predicted period distributions in Fig. 3. The period distribution of LMC stars evolving along the Cepheid instability strip differs from Galactic stars. The LMC binary systems do not appear to interact as often and fewer stars lose a significant fraction of their initial mass. There are more Cepheid binaries with orbital period shorter than one year relative to all systems including mass transfer. By considering those binary Cepheids with orbital period shorter than 1 yr, we find a peak in orbital periods at about 0.4 yr as opposed to between one and two years for Galactic Cepheids. The radius of an LMC star at the tip of the RGB is smaller than for a star with the same mass but with Galactic metallicity suggesting that, for a given separation, a, tides are also weaker. Furthermore, stellar evolution models form Cepheid blue loops at lower mass at smaller metallicity (Pols et al. 1998; Bono et al. 2000a), implying that Cepheid binary systems can evolve to have smaller separations. Binary separations must also be shorter to begin Roche-lobe overflow.
An inclination range between 87 and 89.2 suggests an eclipsing binary fraction of 0.9% and 3.3%, i.e. between 4 and 20 Cepheids observed in the OGLE-III survey should be in eclipsing binary systems. This is consistent with the number presented by Soszynski et al. (2008) of three to seven Cepheid eclipsing binaries.
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