The Physics of RR Lyrae

With periods between 0.2 and 1.0 days, RR Lyrae stars are one of the most useful types of variable stars used for exploring the distances and properties of old stellar populations. In the HR diagram, they are found in the instability strip with absolute visual magnitudes near 0.6 and mean effective temperatures ranging between about 6000 and 7250 K (Catelan 2004). RR Lyrae stars are only found in systems that contain a stellar component older than about 10 Gyr, and they are thus an important standard candle for determining distances to very old systems. The prototype of this class of variable stars, RR Lyrae itself, was discovered by Williamina Fleming on Harvard College Observatory photographs (Pickering et al. 1901). The class of variable stars was then defined through observations of RR Lyrae stars in globular clusters. Between 1895 and 1898, Bailey and Pickering found more than 500 variable stars in a search of 23 globular clusters (Bailey and Pickering 1913). Bailey noticed that many of these variables showed similar

properties: their periods were mostly shorter than a day, and their amplitudes on the blue- sensitive photometric plates were typically about 1 magnitude. These stars were first called cluster variables, of which most are stars we call nowadays RR Lyrae stars, or short, RR Lyrae.

RR Lyrae Types and Light Curve Properties

Bailey (1902) divided the RR Lyrae stars in ω Cen (ω Centauri, NGC 5139) into three sub-classes, now called Bailey types. They can be distinguished by light curve shape, period and amplitude. Bailey type a stars show the largest amplitudes and the steepest rise to the maximum amplitude. RR Lyrae of type b are similar to those of type a, but with smaller amplitudes and longer periods. Type c RR Lyrae have shorter periods and lower amplitudes. Their light curves are more symmetric than those of types a and b, and show an almost sinusoidal shape.

As the type a and b RR Lyrae stars form a continuous sequence in an amplitude-period diagram, it is now usual to combine them into a single type RRac, leaving only the original Bailey type c distinct (RRc). Additionally there is a type called RRd stars, which are double-mode pulsators, unlike RRac or RRc (Nemec 1985). Among all types of RR Lyrae, RRab variables are the most common, making up _{∼91% of all observed RR Lyrae (Smith 2004). RRc variables account for} ∼9% of the observed RR Lyrae, RRd are the rarest RR Lyrae and make up only ∼1% (Smith 2004).

Fig. 2.13 shows light curves of typical RRab and RRc stars. The light curves are given in the ugriz filters used by the SDSS survey (York et al. 2000).

RR Lyrae show an increase in their amplitude as one goes from the near-infrared z filter to the g filter, but with only a small change as one continues to the u filter. Towards the ultraviolet, they reach amplitudes up to 4 magnitudes (Downes et al. 2004). When proceeding to the infrared instead, the decline of the amplitude with increasing wavelength continues.

Whereas some RR Lyrae stars have light curves that repeat nearly perfect from one pulsation cycle to the next, some RR Lyrae stars have light curves that change in a secondary period that can be tens or hundreds of days long. These changes are called the Blazhko effect (Blaˇzko 1907).

RR Lyrae Stars as Standard Candles

RR Lyrae stars are especially important as they can be used to measure distances to systems containing old stellar populations, such as the Milky Way’s halo. They were first used by Harlow Shapley to determine distances to globular clusters, leading to the awareness that the Sun is located far from the center of our Galaxy (Shapley 1914).

Within a single globular cluster, all RR Lyrae have about the same visual apparent magnitude, as they are in the HB stage of the evolution of low-mass stars. This makes RR Lyrae very good standard candles.

Figure 2.13Examples of RRab and RRc light curves from OGLE-II. (a) shows a raw RRab light curve, (c) a raw RRc light curve. In panels (b) and (d), the corresponding outlier removed smoothed light curves are shown. The steep rise in the RRab light curve, in contrast to the almost symmetric nature of the RRc light curve, is clearly visible. Taken from Deb and Singh (2010).

For distance determination, the apparent magnitude must be transformed to absolute magnitudes. Until the 1960s, it was assumed that all RR Lyrae have the same absolute V band magnitude. Later on, a dependency on metallicity was found (Sandage 1990a), that is generally expressed as hMVi = a + b[Fe/H]. A calibration by Benedict et al. (2011) results into the relation

hMVi = (0.214 ± 0.047)([Fe/H] + 1.5) + (4.5 ± 0.05). (2.66)

Longmore et al. (1986) found a linear relationship between the mean infrared K-band magni- tude (λ _{' 2.20 µm) and the logarithm of the RR Lyrae star’s fundamental-mode period. This} infrared period-luminosity relation has the advantage of being relatively insensitive to interstellar

extinction. Also, it is relatively insensitive to the star’s [Fe/H] value. Updated relationships can be found in C´aceres and Catalan (2008), indicating

Mz= 0.839− 1.295 log P + 0.211 log Z (2.67)

Mi= 0.908− 1.035 log P + 0.220 log Z (2.68)

for SDSS i and z bandpasses and metallicity Z, _{P in days.}

The Evolution of RR Lyrae Stars

RR Lyrae are stars who have already left the main sequence (see Fig. 2.9), ascended the RGB, undergone the He flash, and settled down to core He burning that characterizes stars on the HB. It takes more than 10 Gyr until they reach the HB. The lifetime of RR Lyre stars is expected to be in the order of 108 _{years (Koopmann et al. 1994). The variability of RR Lyrae stars is caused by}

pulsation, being mainly driven by κ and γ mechanisms. The zone within the star where He goes from being singly to doubly ionized is most important for driving the pulsations. RRab stars are pulsating in the fundamental radial mode, whereas RRc stars are pulsating in the first-overtone mode.

Period Changes

RR Lyrae can undergo period changes. This came apparent as the time spanned by observations of the same RR Lyrae reached 100 years and more. Whereas some have stable periods, others undergo significant changes. Such period changes have been observed for RR Lyrae in a number of the Milky Way’s globular cluster (Catelan and Smith 2015). It was suggested by Sweigart and Renzini (1979) that discrete mixing events in the semi-convective zone of RR Lyrae could lead to period noise and thus period change. Cox (1998) proposed that small changes in the gradient of the He composition in the regions of RR Lyre stars below the H and He convective zones might produce the period changes.

Period Distributions: The Oosterhoff Groups

Oosterhoff, working on RR Lyrae within 5 globular clusters (Oosterhoff 1939), noted that they could be divided into two groups according their period, now known as the Oosterhoff groups. The globular clusters with a mean period of their RRab _hPabi near 0.55 days became known as

Oosterhoff type I clusters, whereas those with_hPabi near 0.65 days became known as Oosterhoff

type II clusters. Analysis of [Fe/H] showed that globular clusters of Oosterhoff type I are more metal rich than those of Oosterhoff type II. There are various approaches to explain this difference. RR Lyrae stars in Oosterhoff type II clusters are more luminous than those in Oosterhoff type I clusters. The longer periods of Oosterhoff type II RR Lyrae would then be a result of their lower

densities, according to the pulsation equation _{Pphρi = Q. As a possible explanation, Sandage} (1981) suggested that a higher He abundance in the Oosterhoff type II clusters might account for their different period. However, this does not explain why Oosterhoff type II clusters have higher fractions of RRc stars than Oosterhoff type I clusters. Also it does not explain the higher period change rates found in Oosterhoff type II clusters.

A different explanation that accounts for both the higher RRc fraction as well as mean period, was proposed by van Albada and Baker (1973). They suggested the existence of a so-called hysteresis zone near the center of the instability strip. Within this zone, both the fundamental and the first-overtone modes can in principle be excited. RR Lyrae stars entering this zone of the HR diagram would keep the pulsation mode that they had.