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Astro-Ph/9812074 3 Dec 1998 1 Ter Calibrations of Cepheid Luminositites, Alues

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Astro-Ph/9812074 3 Dec 1998 1 Ter Calibrations of Cepheid Luminositites, Alues View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server The Cepheid Distance Scale after Hipparcos Frdric Pont Observatoire de Genve, Switzerland Abstract. More than twohundred classical cepheids were measured by the Hip- parcos astrometric satellite, making p ossible a geometrical calibration of the cepheid distance scale. However, the large average distance of even the nearest cepheids measured by Hipparcos implies trigonometric paral- laxes of at most a few mas. Determining unbiased distances and absolute magnitudes from such high relative error parallax data is not a trivial problem. In 1997, Feast & Catchp ole announced that Hipparcos cepheid paral- laxes indicated a Perio d-Luminosity scale 0.2 mag brighter than previous calibrations, with imp ortant consequences on the whole cosmic distance scale. In the wake of this initial study, several authors have reconsid- ered the question, and favour fainter calibrations of cepheid luminositites, compatible with pre-Hipparcos values. All authors used equivalent data sets, and the bulk of the di erence in the results arises from the statistical treatment of the parallax data. Wehave attempted to rep eat the analyses of all these studies and test them with Monte Carlo simulations and synthetic samples. We conclude that the initial Feast & Catchp ole study is sound, and that the subsequent studies are sub jected in several di erentways to biases involved in the treatment of high relative error parallax data. We consider the source of these biases in some detail. We also prop ose a reappraisal of the error budget in the nal Hipparcos cepheid result, leading to a PL relation { adapted from Feast & Catchp ole { of +0 M = 2:81(assumed) log P 1:43 0:16(stat) (syst) V 0:03 astro-ph/9812074 3 Dec 1998 We compare this calibration to recentvalues from cluster cepheids or the surface brightness metho d, and nd that the overall agreement is good within the uncertainties. We conclude by commenting on the mismatchbetween the cepheid parallax distance scale and kinematical determinations, for cepheids as well as RR Lyrae. 1. Intro duction The cepheid distance scale is the central link of the cosmic distance ladder. Because even the nearest cepheids are more than 100 p c away from the Sun 1 (Polaris at 130 p c, Cep at 300 p c), no reliable parallax measurements were available for cepheids b efore the Hipparcos mission. The zero-p oint of the Perio d-Luminosity (PL) relation was calibrated by secondary metho ds, using cepheids in op en clusters and asso ciations or Baade-Wesselink techniques. The Hipparcos data for cepheids op ened the p ossibility of a geometric determination of the zero-p oint of the PL relation. So on after the Hipparcos data release, Feast & Catchp ole (1997, FC) an- nounced that Hipparcos cepheid data indicated a zero-p oint ab out 0.2 mag brighter than previously thought (implying =18.70 for the LMC). Adopting the FC notation for the PL relation : M = log P + V they nd = 1:43 0:13 (for = 2.81 assumed). This result attracted considerable attention, since it implied a 10% downward revision of the value of H obtained from galaxy recession velo cities. In turn, the higher expansion ages 0 derived could b ecome compatible with the new, lower ages for globular clusters obtained from Hipparcos sub dwarfs, resolving the \cosmic age problem" (the fact that globular cluster ages were found to b e much older than the expansion age of the Universe). In the past year however, several authors reconsidered the Hipparcos cepheid data, and voiced criticism against the FC result. All subsequent studies obtain calibrations for the zero-p oint of the PL relation similar to pre-Hipparcos val- ues, or fainter (see sketch Fig. 1). Szabados (1997) p ointed out that known or susp ected binaries were abundant in the cepheid sample, and that the PL relation found after removing them was compatible with pre-Hipparcos values. Madore & Freedman (1998) rep eated the whole analysis with multi-wavelength data (BVIJHK). They also concluded on a fainter zero-p oint. Oudmaijer et al. (1998), in a study devoted to the e ect of the Lutz-Kelker bias on Hipparcos luminosity calibrations, analyse the cepheid data as an illustration and derive = 1:29 0:02. Finally, Luri et al. (1998) apply the \LM metho d" (Luri et al. 1996) to the cepheid sample and nd a very faint zero-p ointof=1:05 0:17. The situation, one year after the Hipparcos data release, is therefore very p erplexing : while b efore Hipparcos the zero-p oint of the cepheid PL relation seemed reasonably well determined within 0.1 mag (e.g. Gieren et al. 1998), values derived from the same Hipparcos parallax data cover a range of 0.4 mag! Fortunately, this distressing situation may b e only temp orary. As all the studies considered use the same parallax data and similar photometric and red- dening values for Hipparcos cepheids, the di erences are almost entirely due to the statistical pro cedures used in the analyses. Contrarily, for instance, to the case of the very metal-p o or globular cluster distance scale where observational uncertainties may remain the limiting factor (see review by Cacciari in this vol- ume), it may b e hop ed that the disagreements ab out the PL zero-p oint can b e resolved byevaluating the di erent approaches. This is what we are attempting here. From statistical considerations on one hand, Monte Carlo simulations of the di erent pro cedures on the other hand, we have tested the robustness and p ossible biases of the di erent pro cedures. The results are presented b elowinPar. 3 to 8, study by study, b eginning with an 2 Figure 1. Calibrations of the cepheid Perio d-Luminosity relation zero-p oint("") from Hipparcos parallaxes : Feast & Catchp ole (1997), Szabados (1997), Oudmaijer et al. (1998), Madore & Freedman (1998), "direct metho d" (see Par. 3.), and Luri et al. (1998). Typical error bars indicated for one calibration. The dashed area indicates a typical range of pre-Hipparcos values. The PL relation slop e (" ") is xed at 2.81. See also Fig. 5. imaginary author using the unsophisticated "direct metho d". It app ears that the diculties involved in the treatment of high relative error parallax data are at the core of the question. On this sub ject, the review by Arenou in this volume constitute an excellent complement to the present contribution { showing the pitfalls easiest to overlo ok. As far as cepheids are concerned, we shall contend here that, in fact, the situation is rather clear, and that much of the apparent disagreement is caused by statistical biases that have already b een rep orted in dealing with parallax data. 2. Hipparcos cepheids Around 200 cepheids were measured with Hipparcos. All but the nearest ones 1 are so remote that < . There are 19 of them with = < 50%. r eal The closest cepheid, Polaris, was measured at =7:56 0:48 mas. Cep itself has =3:32 0:58 mas. Despite the low accuracy of individual parallax determinations, it is not unreasonable to attempt an accurate determination 1 Parallaxes are noted and their uncertainty 3 of the PL relation zero-p oint, b ecause relative distances are precisely known through the PL and PLC relations. Each individual parallax can therefore b e seen as a measurement of the PL zero-p oint. A large number of low-accuracy parallaxes can yield a reliable combined value of the zero-p oint. 3. Dangers of the \direct metho d" The most natural way to infer absolute magnitude from parallax would b e to calculate distances from the inverse of the parallax and use Pogson's law: m M = 5 log(1= ) 5+a (1) V V V Then, the zero-p oint of the PL relation could b e t in the Perio d-M plane, by V some kind of weighted least-square. In fact, this is an extremely biased way to pro ceed when the relative errors on parallax are high ( = >20%). Fig. 2 shows graphically why the inverse of the parallax is a biased estimator of the distance, and how the magnitude derived from Equ. 1 has a very asymmetrical and skewed distribution if the statistical distribution of the parallax is gaussian. This e ect was p ointed out by several authors in the wake of the publication of the Hipparcos catalogue (e.g. Luri & Arenou 1997). An essential condition to the use of least-square t is the symmetry of the error distribution, and if it is not satis ed, rst-order biases are to b e exp ected. Moreover, in order to use Equ. 1, negative parallaxes must b e ignored and the data selected by some cut in = . Both selection criteria intro duce further biases. By discriminating against low at a given and meas , they bias the result towards lower distances and fainter magnitudes. r eal 02460 500 56789 dist [pc] Figure 2. Statistical distributions of the parallax, distance computed from d =1= and absolute magnitude from Equ. 1 with =3 mag r eal and =1 mas. The distance and absolute magnitude distributions are biased and asymmetrical. Monte Carlo simulations with samples similar to the actual Hipparcos cepheid sample show that this \direct metho d" leads to a bias of 0.2 mag towards a 4 fainter PL relation zero-p oint. An imp ortant bias indeed, due exclusively to the incorrect statistical treatment of parallax data. 4.
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