Period–colour and amplitude–colour relations in classical Cepheid variables – V. The Small Magellanic Cloud Cepheid models

Period–colour (PC) and amplitude–colour (AC) relations at maximum, mean and minimum light are constructed from a large grid of full amplitude hydrodynamic models of Cepheids with a composition appropriate for the Small Magellanic Cloud (SMC). We compare these theoretical relations with those from observations. The theoretical relations are, in general, in good agreement with their observational counterparts, though there exist some discrepancy for short period  (log [ P ] < 1)  Cepheids. We outline a physical mechanism which can, in principle, be one factor to explain the observed PC/AC relations for the long and short period Cepheids in the Galaxy, Large Magellanic Cloud (LMC) and SMC. Our explanation relies on the hydrogen ionization front (HIF)–photosphere interaction and the way this interaction changes with pulsation period, pulsation phase and metallicity. Since the PC relation is connected with the period–luminosity (PL) relation, it is postulated that such a mechanism can also explain the observed properties of the PL relation in these three galaxies.     

Period–colour and amplitude–colour relations in classical Cepheid variables – III. The Large Magellanic Cloud Cepheid models

Period–colour (PC) and amplitude–colour (AC) relations are studied for the Large Magellanic Cloud (LMC) Cepheids under the theoretical framework of the hydrogen ionization front (HIF)–photosphere interaction. LMC models are constructed with pulsation codes that include turbulent convection, and the properties of these models are studied at maximum, mean and minimum light. As with Galactic models, at maximum light the photosphere is located next to the HIF for the LMC models. However, very different behaviour is found at minimum light. The long-period  ( P > 10 d)  LMC models imply that the photosphere is disengaged from the HIF at minimum light, similar to the Galactic models, but there are some indications that the photosphere is located near the HIF for the short-period  ( P < 10 d)  LMC models. We also use the updated LMC data to derive empirical PC and AC relations at these phases. Our numerical models are broadly consistent with our theory and the observed data, though we discuss some caveats in the paper. We apply the idea of the HIF–photosphere interaction to explain recent suggestions that the LMC period–luminosity (PL) and PC relations are non-linear with a break at a period close to 10 d. Our empirical LMC PC and PL relations are also found to be non-linear with the F -test. Our explanation relies on the properties of the Saha ionization equation, the HIF–photosphere interaction and the way this interaction changes with the phase of pulsation and metallicity to produce the observed changes in the LMC PC and PL relations.     

Period–luminosity relations for type II Cepheids and their application

JHK _s magnitudes corrected to mean intensity are estimated for Large Magellanic Cloud (LMC) type II Cepheids in the OGLE-III survey the third phase of the Optical Gravitational Lensing Experiment (OGLE). Period–luminosity (PL) relations are derived in JHK _s as well as in a reddening-free VI parameter. Within the uncertainties, the BL Her stars  ( P < 4 d)  and the W Vir stars (   P = 4  to 20 d) are colinear in these PL relations. The slopes of the infrared relations agree with those found previously for type II Cepheids in globular clusters within the uncertainties. Using the pulsation parallaxes of V553 Cen and SW Tau, the data lead to an LMC modulus uncorrected for any metallicity effects of  18.46 ± 0.10  mag. The type II Cepheids in the second-parameter globular cluster, NGC 6441, show a PL( VI ) relation of the same slope as that in the LMC, and this leads to a cluster distance modulus of  15.46 ± 0.11  mag, confirming the hypothesis that the RR Lyrae variables in this cluster are overluminous for their metallicity. It is suggested that the Galactic variable κ Pavonis is a member of the peculiar W Vir class found by the OGLE-III group in the LMC. Low-resolution spectra of OGLE-III type II Cepheids with   P > 20  d (RV Tau stars) show that a high proportion have TiO bands; only one has been found showing C₂. The LMC RV Tau stars, as a group, are not colinear with the shorter period type II Cepheids in the infrared PL relations in marked contrast to such stars in globular clusters. Other differences between LMC, globular cluster and Galactic field type II Cepheids are noted in period distribution and infrared colours.     

Period–luminosity relations for red supergiant variables – I.The calibration

We present CCD photometry of red supergiant long-period variables (LPVs) in the Per OB1 association, the Large Magellanic Cloud (LMC) and M33. The photometry was obtained in the Kron–Cousins R and I bandpasses and in a narrow bandpass ( λ ₀=8250 Å, FWHM=300 Å) chosen to avoid TiO bands in the spectral energy distribution of the LPVs. Because the strength of the TiO bands varies greatly with temperature, which varies with the phase of an LPV, avoiding TiO reduces the amplitude of the photometric variations seen in LPVs. The result is a lower dispersion and a well defined period–luminosity (PL) relation.
For the LMC sample we find an rms dispersion of 0.27 mag in the narrow-band PL relation and slightly larger dispersions for the LPVs in Per OB1 and M33. This dispersion is comparable to that of the Cepheid PL relation at similar wavelengths. Adopting a distance modulus of 18.5±0.1 mag for the LMC, we obtain distance moduli of 11.68±0.15 mag for Per OB1 and 24.85±0.13 mag for M33. These distances agree well with those based on main sequence fitting for Per OB1 and the Cepheid distance for M33. Since LPVs are ∼ 5 times more common than Cepheids and have a well defined PL relation, LPVs provide a promising method for estimating Galactic and extra galactic distances.     

The linearity of the Wesenheit function for the Large Magellanic Cloud Cepheids

There is strong evidence that the period–luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break at a period around 10 d. Because the LMC PL relation is extensively used in distance scale studies, the non-linearity of the LMC PL relation may affect the results based on this LMC calibrated relation. In this paper we show that this problem can be remedied by using the Wesenheit function in obtaining Cepheid distances. This is because the Wesenheit function is linear, although recent data suggest that the PL and the period–colour (PC) relations that make up the Wesenheit function are not. We test the linearity of the Wesenheit function and find strong evidence that the LMC Wesenheit function is indeed linear. This is because the non-linearity of the PL and PC relations cancel out when the Wesenheit function is constructed. We discuss this result in the context of distance scale applications. We also compare the distance moduli obtained from  μ₀=μ _V − R (μ _V −μ _I )  (equivalent to Wesenheit functions) constructed with the linear and the broken LMC PL relations, and we find that the typical difference in distance moduli is  ∼ ±0.03 mag  . Hence, the broken LMC PL relation does not seriously affect current distance scale applications. We also discuss the random error calculated with equation  μ₀=μ _V − R (μ _V −μ _I )  , and show that there is a correlation term that exists from the calculation of the random error. The calculated random error will be larger if this correlation term is ignored.     

AGB variables and the Mira period–luminosity relation

Published data for large-amplitude asymptotic giant branch variables in the Large Magellanic Cloud (LMC) are re-analysed to establish the constants for an infrared ( K ) period–luminosity relation of the form   M _K =ρ[log  P − 2.38]+δ  . A slope of  ρ=−3.51 ± 0.20  and a zero-point of  δ=−7.15 ± 0.06  are found for oxygen-rich Miras (if a distance modulus of 18.39 ± 0.05 is used for the LMC). Assuming this slope is applicable to Galactic Miras we discuss the zero-point for these stars using the revised Hipparcos parallaxes together with published very long baseline interferometry (VLBI) parallaxes for OH masers and Miras in globular clusters. These result in a mean zero-point of  δ=−7.25 ± 0.07  for O-rich Galactic Miras. The zero-point for Miras in the Galactic bulge is not significantly different from this value.
Carbon-rich stars are also discussed and provide results that are consistent with the above numbers, but with higher uncertainties. Within the uncertainties there is no evidence for a significant difference between the period–luminosity relation zero-points for systems with different metallicity.     

Further empirical evidence for the non-linearity of the period–luminosity relations as seen in the Large Magellanic Cloud Cepheids

Recent studies, using OGLE data for LMC Cepheids in the optical, strongly suggest that the period–luminosity (PL) relation for the Large Magellanic Cloud (LMC) Cepheids shows a break or non-linearity at a period of 10 d. In this paper we apply statistical tests, the chi-squared test and the F -test, to the Cepheid data from the MACHO project to test for a non-linearity of the V - and R -band PL relations at 10 d, and extend these tests to the near-infrared ( JHK -band) PL relations with 2MASS data. We correct the extinction for these data by applying an extinction map towards the LMC. The statistical test we use, the F -test, is able to take account of small numbers of data points and the nature of that data on either side of the period cut at 10 d. With our data, the results we obtained imply that the VRJH -band PL relations are non-linear around a period of 10 d, while the K -band PL relation is (marginally) consistent with a single-line regression. The choice of a period of 10 d, around which this non-linearity occurs, is consistent with the results obtained when this 'break' period is estimated from the data. We show that robust parametric (including least-squares, least absolute deviation, robust regression) and non-parametric regression methods, which restrict the influence of outliers, produce similar results. Long-period Cepheids are supplemented from the literature to increase our sample size. The photometry of these long-period Cepheids is compared with our data and no trend with period is found. Our main results remain unchanged when we supplement our data set with these long-period Cepheids. By examining our data at maximum light, we also suggest arguments as to why errors in reddening are unlikely to be responsible for our results. The non-linearity of the mean V -band PL relation as seen in both of the OGLE and the MACHO data, using different extinction maps, suggests that this non-linearity is real.     

The period–luminosity relation for type II Cepheids in globular clusters

We report the result of our near-infrared observations ( JHK _s) for type II Cepheids (including possible RV Tau stars) in galactic globular clusters. We detected variations of 46 variables in 26 clusters (10 new discoveries in seven clusters) and present their light curves. Their periods range from 1.2 d to over 80 d. They show a well-defined period–luminosity relation at each wavelength. Two type II Cepheids in NGC 6441 also obey the relation if we assume the horizontal branch stars in NGC 6441 are as bright as those in metal-poor globular clusters in spite of the high metallicity of the cluster. This result supports the high luminosity which has been suggested for the RR Lyr variables in this cluster. The period–luminosity relation can be reproduced using the pulsation equation     assuming that all the stars have the same mass. Cluster RR Lyr variables were found to lie on an extrapolation of the period–luminosity relation. These results provide important constraints on the parameters of the variable stars.
Using Two Micron All-Sky Survey (2MASS) data, we show that the type II Cepheids in the Large Magellanic Cloud (LMC) fit our period–luminosity relation within the expected scatter at the shorter periods. However, at long periods (   P > 40  d, i.e. in the RV Tau star range) the LMC field variables are brighter by about one magnitude than those of similar periods in galactic globular clusters. The long-period cluster stars also differ from both these LMC stars and galactic field RV Tau stars in a colour–colour diagram. The reasons for these differences are discussed.     

Cepheid parallaxes and the Hubble constant

Revised Hipparcos parallaxes for classical Cepheids are analysed together with 10 Hubble Space Telescope ( HST )-based parallaxes. In a reddening-free V , I relation we find that the coefficient of log  P is the same within the uncertainties in our Galaxy as in the Large Magellanic Cloud (LMC), contrary to some previous suggestions. Cepheids in the inner region of NGC 4258 with near solar metallicities confirm this result. We obtain a zero-point for the reddening-free relation and apply it to the Cepheids in galaxies used by Sandage et al. to calibrate the absolute magnitudes of Type Ia supernova (SNIa) and to derive the Hubble constant. We revise their result for H ₀ from 62 to 70 ± 5 km s⁻¹ Mpc⁻¹. The Freedman et al. value is revised from 72 to 76 ± 8 km s⁻¹ Mpc⁻¹. These results are insensitive to Cepheid metallicity corrections. The Cepheids in the inner region of NGC 4258 yield a modulus of 29.22 ± 0.03 (int.) compared with a maser-based modulus of 29.29 ± 0.15. Distance moduli for the LMC, uncorrected for any metallicity effects, are 18.52 ± 0.03 from a reddening-free relation in V , I ; 18.47 ± 0.03 from a period–luminosity relation at K ; 18.45 ± 0.04 from a period–luminosity–colour relation in J , K . Adopting a metallicity correction in V , I from Macri et al. leads to a true LMC modulus of 18.39 ± 0.05.     

Carbon-rich Mira variables: kinematics and absolute magnitudes

The kinematics of Galactic C-Miras are discussed on the basis of the bolometric magnitudes and radial velocities of Papers I and II of this series. Differential Galactic rotation is used to derive a zero-point for the bolometric period–luminosity relation which is in satisfactory agreement with that inferred from the Large Magellanic Cloud (LMC) C-Miras. We find for the Galactic Miras,   M _bol=−2.54 log  P + 2.06(±0.24)  , where the slope is taken from the LMC. The mean velocity dispersion, together with the data of Nordström et al. and the Padova models, leads to a mean age for our sample of C-Miras of  1.8 ± 0.4 Gyr  and a mean initial mass of  1.8 ± 0.2 M_⊙  . Evidence for a variation of velocity dispersion with period is found, indicating a dependence of period on age and initial mass, the longer period stars being younger. We discuss the relation between the O- and C-Miras and also their relative numbers in different systems.     

Globular clusters and the Mira period–luminosity relation

A globular cluster distance scale based on Hipparcos parallaxes of subdwarfs has been used to derive estimates of M _K for cluster Miras, including one in the Small Magellanic Cloud (SMC) globular cluster NGC 121. These lead to a zero-point of the Mira infrared period–luminosity (PL) relation, PL( K ), in good agreement with that derived from Hipparcos parallaxes of nearby field Miras. The mean of these two estimates together with data on LMC Miras yields a Large Magellanic Cloud (LMC) distance modulus of     in evident agreement with a metallicity-corrected Cepheid modulus     .
The use of luminous asymptotic giant branch (AGB) stars as extragalactic population indicators is also discussed.     

Confirmation of previous ground-based Cepheid P–L zero-points using Hipparcos trigonometric parallaxes

Comparisons show agreement at the 0.1-mag level between the calibration of the Cepheid period–luminosity (P–L) relation by Feast & Catchpole (FC) using the early release of Hipparcos data and four previous ground-based calibrations, three of which are either largely or totally independent of the distance to the Large Magellanic Cloud (LMC). Each of the comparisons has the sense that the FC calibration is brighter, but only at the level of ≲0.1 mag. In contrast, FC argue that their Hipparcos recalibration leads to a 0.2-mag revision in the distance to the LMC, and thereby to a 10 per cent decrease in the Hubble constant. We argue differently. The comparison of the Hipparcos recalibration with others should be made using only local Galactic Cepheids, not based on Cepheids in the LMC that require a set of precepts that are not germane to the direct Hipparcos recalibration. The comparison made here, using only Galactic Cepheids, gives a correction of ∼4 per cent or less to our value of H ₀ based on Type Ia supernovae, keeping all other factors and precepts the same.
  A second success of the Hipparcos mission is the calibration of the position of the main sequence in the Hertzsprung–Russell diagram as a function of metallicity using local subdwarfs. These data have been used by Reid and by Gratton et al. to obtain, similarly to FC, a brighter absolute magnitude of RR Lyrae stars by ∼0.3 mag from that often currently adopted. These new calibrations confirm the earlier brighter calibrations by Walker, by Sandage, and by Mazzitelli, D'Antona & Caloi, thereby reducing the ages of globular clusters by ∼30 per cent. This removes most of the cosmological time-scale problem if H ₀∼55 km s⁻¹ Mpc⁻¹. A similar conclusion, based on pulsation theory and MACHO data, has been reached by Alcock et al.     

Hipparcos parallaxes for Mira-like long-period variables

This paper concerns the calibration of the K period–luminosity relation for Mira variables using Hipparcos parallaxes. K magnitudes are available for 255 Mira-like variables which were observed by Hipparcos . Period–luminosity zero-points are evaluated for various subgroups of data. The best solution for oxygen-rich Miras, which uses 180 stars, omitting the short-period red group (which had different kinematics from the short-period blue stars) and the low-amplitude variables, provides a zero-point of     which implies a distance modulus for the Large Magellanic Cloud of     or perhaps slightly greater if a metallicity correction is required, in good agreement with the value derived from Cepheids. The zero-point of the period–luminosity relation for carbon stars is briefly discussed.
Linear diameters are derived for red variables with measured angular diameters and parallaxes, and are used to examine the long-standing question of the pulsation mode(s) of these stars. Evidence is presented to suggest that most of them are pulsating in the same mode and, if published model atmospheres are correct, this is probably the first overtone. Some discussion is given of sequences in the period–luminosity and period–colour diagrams and their bearing on the pulsation mode problem.     

On the second-overtone stability among Small Magellanic Cloud Cepheids

We present a new set of Cepheid, full amplitude, non-linear, convective models which are pulsationally unstable in the second overtone (SO). Hydrodynamical models were constructed by adopting a chemical composition typical of Cepheids in the Small Magellanic Cloud (SMC) and for stellar masses ranging from 3.25 to 4 M_⊙. Predicted φ ₂₁ Fourier parameters agree, within current uncertainties, with empirical data for pure first- and second-overtone variables as well as for first-/second-overtone (FO/SO) double-mode Cepheids collected by Udalski et al. in the SMC. On the other hand, predicted I -band amplitudes are systematically larger than the observed ones in the short-period range, but attain values that are closer to the empirical ones for log  P _SO≥−0.12 and log  P _FO≥0.1 . We also find, in agreement with empirical evidence, that the region within which both second and first overtones attain a stable limit cycle widens when moving towards lower luminosities. Moreover, predicted P _SO/ P _FO and P _FO/ P _F period ratios agree quite well with empirical period ratios for FO/SO and fundamental/FO double-mode SMC Cepheids.
Interestingly enough, current models support the evidence that pure SO Cepheids and SO components in FO/SO Cepheids are good distance indicators. In fact, we find that the fit of the predicted period–luminosity–colour ( V , V – I ) relation to empirical SMC data supplies a distance modulus ( DM ) of 19.11±0.08 mag . The same outcome applies to pure FO Cepheids and FO components in FO/SO Cepheids, and indeed we find DM =19.16±0.19 mag . Current distance estimates do not account for, within current uncertainties on photometry and reddening, the so-called short distance scale.     

Period–magnitude relations for M giants in Baade's Window NGC 6522

A large and complete sample of stars with   K < 9.75  in the NGC 6522 Baade's Window towards the Galactic bulge is examined using light curves extracted from MACHO and IJK photometry from DENIS.
The improved statistics resulting from a sample of over 1000 variables allow all four of the sequences A, B, C and D in the   K _S   , log  P diagram of the Large Magellanic Cloud to be seen in the Galactic bulge. The bulge sequences, however, show some differences from those in the Magellanic Clouds, possibly due to the effects of higher metallicity. These sequences may have value as distance indicators.
A new diagram of the frequency of late-type variables at a given amplitude is derived and compared with the old one from Payne-Gaposchkin. The catalogued semiregular variables of the solar neighbourhood are found to be a subset of the total of semiregulars, biased towards large amplitude.     

The kinematics and zero-point of the log P–〈MK〉 relation for Galactic RR Lyrae variables via statistical parallax

We use accurate absolute proper motions and Two-Micron All-Sky Survey   K_s   -band apparent magnitudes for 364 Galactic RR Lyrae variables to determine the kinematical parameters of the Galactic RR Lyrae population and constrain the zero-point of the   K_s   -band period–luminosity relation for these stars via statistical parallax. We find the mean velocities of the halo- and thick-disc RR Lyrae populations in the solar neighbourhood to be  [ U ₀(Halo), V ₀(Halo), W ₀(Halo)]= (−12 ± 10, −217 ± 9, −6 ± 6) km s⁻¹  and  [ U ₀(Disc), V ₀(Disc), W ₀(Disc)]= (−15 ± 7, −44 ± 7, −25 ± 5) km s⁻¹  , respectively, and the corresponding components of the velocity-dispersion ellipsoids,  [σ V_R (Halo), σ V _θ(Halo), σ W (Halo)]= (167 ± 9, 86 ± 6, 78 ± 5) km s⁻¹  and  [σ V_R (Disc), σ V _θ(Disc), σ W (Disc)]= (55 ± 7, 44 ± 6, 30 ± 4) km s⁻¹  , respectively. The fraction of thick-disc stars is estimated at  0.25 ± 0.03  . The corrected infrared period–luminosity relation is     , implying a Large Magellanic Cloud (LMC) distance modulus of  18.27 ± 0.08  and a solar Galactocentric distance of  7.58 ± 0.40 kpc  . Our results suggest no or slightly prograde rotation for the population of halo RR Lyraes in the Milky Way.     

Calibrating the Cepheid Period-Luminosity relation from the near-infrared surface brightness technique

We have applied the near-infrared surface-brightness method to 111 Cepheids in the Milky Way and in the Large and the Small Magellanic Clouds determining distances and luminosities for the individual stars. We find that the K-band Period-Luminosity (PL-)relations for Milky Way and Large Magellanic Cloud Cepheids are almost identical, whereas the zero point of the Wesenheit relation depends significantly on metallicity, metal poor Cepheids being fainter.     

The absolute magnitudes of Cepheids and the extragalacticdistance scale

It is pointed out that a Cepheid period–luminosity relation with a zero-point from Hipparcos trigonometrical parallaxes and a consistent reddening system zero-point implies that some recent estimates of H ₀ based on the Cepheid scale should be increased by ∼8 per cent. This result avoids using the distance to the Large Magellanic Cloud (LMC) as an intermediary point but is not significantly different from the result obtained by Feast & Catchpole via the LMC. A number of other issues are discussed, including metallicity effects on Cepheid distances and reddenings, and the age of metal-poor globular clusters.     

Extinction of H ii regions in the Large Magellanic Cloud

The extinction properties of H  ii regions in the Large Magellanic Cloud are investigated using radio continuum data obtained from the Molonglo Observatory Synthesis Telescope, digitized and calibrated Hα data and published Balmer decrement measurements. The resulting extinction–colour excess diagram suggests that (1) most H  ii regions in the Magellanic Clouds have similar extinction properties to the Galactic ones, (2) all imaginable gas/dust configurations are possible, (3) the extinction of some highly reddened H  ii region cores originates externally in cocoon shells.   The puzzle of different extinction–colour excess ratios of Galactic and extragalactic H  ii regions is explained as being due to the different populations of observed samples rather than any intrinsic differences. The extinction of the observed Galactic H  ii regions produced by foreground dust overwhelms the internal extinction, while the situation in the observed extragalactic H  ii regions is just the opposite.     

Direct calibration of the Cepheid period–luminosity relation

After the first release of Hipparcos data, Feast & Catchpole gave a new value for the zero-point of the visual Cepheid period–luminosity relation, based on trigonometric parallaxes. Because of the large uncertainties on these parallaxes, the way in which individual measurements are weighted is of crucial importance. We therefore conclude that the choice of the best weighting system can be aided by a Monte Carlo simulation. On the basis of such a simulation, it is shown that (i) a cut ‐off in π or in σ _π π introduces a strong bias; (ii) the zero-point is more stable when only the brightest Cepheids are used; and (iii) the Feast & Catchpole weighting gives the best zero-point and the lowest dispersion. After correction, the adopted visual period–luminosity relation is 〈 M _V 〉=−2.77 log  P −1.44±0.05. Moreover, we extend this study to the photometric I band (Cousins) and obtain 〈 M _I 〉=−3.05 log  P −1.81±0.09.