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Harmonizing the RR Lyrae and Clump Distance Scales - Stretching the Short Distance Scale to Intermediate Ranges?

I explore the consequences of making the RR Lyrae and clump giant distance scales consistent in the solar neighborhood, Galactic bulge and Large Magellanic Cloud (LMC). I employ two major assumptions: 1) that the absolute magnitude - metallicity, M_V(RR) - [Fe/H], relation for RR Lyrae stars is universal, and 2) that absolute I-magnitudes of clump giants, M_I(RC), in Baade's Window are known (e.g., can be inferred from the local Hipparcos-based calibration or theoretical modeling). A comparison between the solar neighborhood and Baade's Window sets M_V(RR) at [Fe/H] = -1.6 in the range (0.59 +/- 0.05, 0.70 +/- 0.05), somewhat brighter than the statistical parallax solution. More luminous RR Lyrae stars imply younger ages of globular cluster, which would be in better agreement with the conclusions from the currently favored stellar evolution and cosmological models. A comparison between Baade's Window and the LMC sets the M_I^{LMC}(RC) in the range (-0.33 +/- 0.09, -0.53 +/- 0.09). The distance modulus to the LMC, mu^{LMC}, is between 18.24 +/- 0.08 and 18.44 +/- 0.07. Unlike M_I^{LMC}(RC), this range in mu^{LMC} does NOT depend on the adopted value of the dereddened LMC clump magnitude, I_0^{LMC}(RC). I argue that the currently available information is insufficient to select the correct distance scale with high confidence.


Introduction
The Hubble Space Telescope Key Project (e.g., Madore et al. 1999) concluded that the biggest uncertainty in the Hubble constant, H 0 , comes from the uncertainty in the distance to the LMC. Among the major methods that have been used to determine the distance to the LMC are: the echo of the supernova 1987A, solving parameters of eclipsing binaries, Cepheids, RR Lyrae stars, and red clump giants. They all suffer from some uncertainties and possible systematic errors. The echo of the supernova 1987A was a transient event with limited data and contradictory interpretations (Gould & Uza 1998versus Panagia 1998. Only one attempt of solving eclipsing binary using space-based spectra was made by Guinan et al. (1998) for HV 2274. Their result is sensitive to the reddening toward HV 2274 (Udalski et al. 1998versus Nelson et al. 2000. To be calibrated with high precision, Cepheids have to wait for the next generation astrometric missions (for the Hipparcos-based calibration see Feast &Catchpole 1997 andPont 1999). The absolute V -magnitudes of RR Lyrae stars, M V (RR), are still under debate with a faint value given by the statistical parallax method and a bright value suggested by the main sequence fitting (see Popowski & Gould 1999). The major problem of the red clump method is the possibility that the absolute I-magnitude, M I (RC), is sensitive to the environment (Cole 1998;Girardi et al. 1998;Twarog, Anthony-Twarog, & Bricker 1999). The mentioned methods give results inconsistent within their estimated uncertainties, which suggests hidden systematics.
Here I concentrate on two horizontal-branch standard candles: red clump giants and RR Lyrae stars. I start with a very short review of their application to determine the distance to the LMC. Paczyński & Stanek (1998) pointed out that clump giants should constitute an accurate distance indicator. In a study of the morphology of the red clump, Beaulieu & Sackett (1998) argued that a distance modulus of µ LMC = 18.3 provides the best fit to the dereddened LMC color-magnitude diagram. Udalski et al. (1998a) and Stanek, Zaritsky, & Harris (1998) applied the I-magnitude based approach of Paczyński and Stanek (1998) and found a very short distance to the LMC (µ LM C ≈ 18.1). In response, Cole (1998) and Girardi et al. (1998) suggested that clump giants are not standard candles and that their M I (RC) depend on the metallicity and age of the population. Udalski (1998bUdalski ( , 1998c countered this criticism by showing that the metallicity dependence is at a low level of about 0.1 mag/dex, and that the M I (RC) is approximately constant for cluster ages between 2 and 10 Gyr. The new determinations of the M I (RC) -[Fe/H] relation by Stanek et al. (2000), Udalski (2000) and Popowski (2000) indicate a moderate slope of 0.10 − 0.20 mag/dex. The only clump determination, which results in a truly long distance to the LMC is a study by Romaniello et al. (2000) who investigated the field around supernova SN 1987A, which is not well suited for extinction determinations. Romaniello et al. (2000) also assumed a bright M I (RC) from theoretical models. To address the issue of possible extinction overestimate in earlier studies (see e.g., Zaritsky 1999 for a discussion), Udalski (1998cUdalski ( , 2000 measured clump magnitudes in low extinction regions in and around the LMC clusters. The resulting µ LM C = 18.24 ± 0.08 (Udalski 2000) is often perceived as the least model-dependent distance modulus to the LMC obtained from clump giants. Different methods to determine the RR Lyrae absolute magnitude are analyzed in Popowski & Gould (1999). The results depend on the methods used. When the kinematic or geometric determinations are employed, one obtains M V (RR) = 0.71 ± 0.07 at [Fe/H] = −1.6 (with M V (RR) = 0.77 ± 0.13 from the best understood method, statistical parallax). The other methods typically produce or are consistent with brighter values. The representative main sequence fitting to globular clusters gives M V (RR) = 0.45 ± 0.12 at [Fe/H] = −1.6 (Carretta et al. 2000). When coupled with Walker (1992) photometry of globular clusters, Popowski & Gould's (1999) best M V (RR) results in µ LM C = 18.33 ± 0.08. When Udalski et al. (1999) photometry of the LMC field RR Lyrae stars is used, one obtains µ LM C = 18.23 ± 0.08.
The essence of the approach presented here is a comparison between clump giants and RR Lyrae stars in different environments. If answers from two distance indicators agree then either the systematics have been reduced to negligible levels in both of them or the biases conspire to produce the same answer. This last problem can be tested with an attempt to synchronize distance scales in three different environments, because a conspiracy of systematic errors is not likely to repeat in all environments. Here I show that combining the information on RR Lyrae and red clump stars in the solar neighborhood, Galactic bulge, and LMC provides additional constraints on the local distance scale.

Assumptions and Observational Data
The results I present in §3 and §4 are not entirely general and have been obtained based on certain theoretical assumptions about the nature of standard candles and populations in different stellar systems. In addition, the conclusions depend on the source of photometry. One does not have much freedom in this regard, but I have made certain choices, which I describe in §2.2.

Theoretical assumptions
This investigation relies strongly on the following two assumptions: 1. The M V (RR) -[Fe/H] relation for RR Lyrae stars is universal. More specifically, I assume that for every considered system, M V (RR) is only a linear function of this system's metallicity: Moreover, I will assume that the slope α = 0.18 ± 0.03, which is not critical for the method but determines the numerical results. In the most general case, M V (RR) depends on morphology of the horizontal branch (Lee, Demarque, & Zinn 1990;Caputo et al. 1993). However, for average non-extreme environments (here the character of environment can be judged using the Lee 1989 index) a linear, universal M V (RR) -[Fe/H] should be a reasonable description. For the RR Lyrae stars of the Galactic halo (either in the solar neighborhood or in Baade's Window) and of the LMC field or globular clusters, equation (1) with universal α and β should approximately hold. The universal character of the calibration is essential to any distance determination with standard candles, and so this assumption is rather standard.
2. The absolute magnitude M BW I (RC) of the bulge clump giants is known, which in practice means one of two things: either one takes the results of population modeling or infers the value from the Hipparcos-calibrated M HIP I (RC) of the local clump stars. I will temporarily adopt the second route and assume that there are no population factors except metallicity that influence M BW I (RC) in the Galactic bulge (with respect to the local clump) or that their contributions cancel out. Again, this is somewhat similar to point 1., but here I am more flexible allowing M LMC I (RC) in the LMC not to follow the local Hipparcos calibration (that is, I allow population effects of all types).

Data
The calibration of clump giants in the solar neighborhood is based on Hipparcos (Perryman 1997) data for nearly 300 clump giants as reported by Stanek & Garnavich (1998) and refined by Udalski (2000).  Table 1. Equation (2) and the following considerations serve only as the evidence that, in the lack of significant population effects, this choice of M BW I (RC) would be well justified.
The V -and I-band photometry for the bulge clump giants and RR Lyrae stars originates from, or have been calibrated to the photometric zero-points of, phase-II of the Optical Gravitational Lensing Experiment (OGLE). That is, the data for Baade's Window come from Udalski (1998b) and were adjusted according to zero-point corrections given by Paczyński et al. (1999). When taken at face value, these data result in (V − I) 0 colors 1 of both clump giant and RR Lyrae stars that are 0.11 redder than for their local counterparts. To further describe the input data let me define ∆ for a given stellar system as the difference between the mean dereddened I-magnitude of clump giants and the derredened V-magnitude of RR Lyrae stars at the metallicity of RR Lyrae stars in the Galactic bulge. The quantity ∆ allows one to compare the relative brightness of clump giants and RR Lyrae stars in different environments and so will be very useful for this study (for more discussion see Udalski 1998b andPopowski 2000). In the Baade's Window with anomalous horizontal branch colors ∆ BW ≡ I BW 0 (RC) − V BW 0 (RR) = −1.04 ± 0.04. When the color correction considered by Popowski (2000) is taken into account one obtains ∆ BW = −0.93 ± 0.04.
In the LMC, I use dereddened I 0 = 17.91 ± 0.05 for "representative red clump". Here "representative" means in clusters (compare to I 0 = 17.88 ± 0.05 from Udalski 1998c) or in fields around clusters (compare to I 0 = 17.94 ± 0.05 from Udalski 2000). The advantage of using I 0 from cluster and cluster fields is their low, well-controlled extinction (Udalski 1998c(Udalski , 2000. I take V 0 = 18.94 ± 0.04 for field RR Lyrae stars at [Fe/H] = −1.6 from Udalski et al. (1999) and adopt V 0 = 18.98 ± 0.03 at [Fe/H] = −1.9 for the cluster RR Lyrae stars investigated by Walker (1992). The difference of photometry between Udalski et al. (1999) and Walker (1992) may have several sources. The least likely is that the cluster system is displaced with respect to the center of mass of the LMC field. Also, cluster RR Lyrae stars could be intrinsically fainter, but again this is not very probable. I conclude that the difference comes either from 1) extinction, or 2) the zero-points of photometry. The first case would probably point to overestimation of extinction by OGLE, because it is harder to determine the exact extinction in the field than it is in the clusters. The second case can be tested with independent LMC photometry. In any case, the difference of ∼ 0.1 mag is an indication of how well we currently measure V 0 (RR) in the LMC.
Finally, let us note that the homogeneity of photometric data was absolutely essential for the investigation of the global slope in the M I (RC) -[Fe/H] relation (Popowski 2000).
Here it is not as critical. Still, the common source of data for the Galactic bulge reduces the uncertainty in the M V (RR) calibration. On the other hand, the use of both OGLE and Walker's (1992) data for the LMC quantifies a possible level of extinction/photometry uncertainty.

The method and results
The distance modulus to the Galactic center from RR Lyrae stars is: I assume the RR Lyrae metallicities of [Fe/H] BW RR = −1.0 from Walker & Terndrup (1991). The distance modulus to the Galactic center from the red clump can be expressed as: The condition that µ BW (RR) and µ BW (RC) are equal to each other 2 results in: But the right hand side of equation (5) is just ∆ BW , which is either directly taken from dereddened data or determined by solving the color problem (for more detail see Popowski 2000). If there are no population differences between the clump in Baade's Window and the solar neighborhood (as we assumed in §2.1), then M BW I (RC) is extremely well constrained from the Hipparcos results reported in equation (2). Therefore, equation (5)  To determine M LMC I (RC), I construct the Udalski's (1998b) diagram. However, both Udalski (1998b) and Popowski (2000) used such diagrams to determine a global slope of the M I (RC) -[Fe/H] relation. Because I am interested here just in the LMC, a more powerful approach is to treat the Udalski (1998b) diagram in a discrete way. That is, instead of fitting a line to a few points one takes a difference between the Baade's Window and LMC ∆ as a measure of the M I (RC) differences in these two stellar systems. Therefore: The interesting feature of equation (7) (6) and so does not provide any additional information.
Using equations (6) and (7) Table  1. Different assumptions about the color anomaly in the Galactic bulge and the use of either OGLE-II or Walker's (1992) where the three δ-type terms indicate potential corrections, α true is a real slope in RR Lyrae M V (RR) -[Fe/H] relation and M BW I,true (RC) is a true absolute magnitude of clump giants in the Bulge. The multiplying factor of 0.6 in the first term is a difference between the solar neighborhood and Baade's Window metallicity of RR Lyrae stars. The distance scale could be made longer with either a larger (steeper) slope α true or a brighter M BW I,true (RC) value. Very few M V (RR) -[Fe/H] relation determinations argue for slopes steeper than 0.3, and clump giants in the Galactic bulge, which are old, are expected to be on average somewhat fainter than the ones in the solar neighborhood. To give an example of application of equation (8) Table 4 without any adjustment for a small [Fe/H] mismatch). The first term would result in a correction of −0.07 mag and the second term would contribute 0.08 mag. In this case the two corrections would almost entirely cancel out resulting in both β and M LMC I (RC) being 0.01 mag fainter and µ LMC being 0.01 mag smaller. Even if one ignores the M BW I,true (RC) -related correction, it is hard to make absolute magnitudes of RR Lyrae and clump stars brighter by more than 0.07 mag. Consequently, the distance moduli to the LMC reported in Table 1 are unlikely to increase by more than 0.07 mag as a result of adjustment to the theoretical assumptions from §2.
Another interesting question is the sensitivity of the results reported in Table 1 to the deredenned magnitudes adopted for the LMC. These dependences are described by the following equations: where V LMC 0 (RR) is either Udalski et al. (1999) or Walker (1992) value described in §2.2. In this treatment, the obtained distance modulus to the LMC does not depend on the dereddened I-magnitudes of clump giants! This is very fortunate because of the unresolved observational controversy [I LMC 0 (RC) ∼ 17.9 from Udalski (1998cUdalski ( , 2000 versus I LMC 0 (RC) ∼ 18.1 from Zaritsky (1999) or Romaniello et al. (1999)]. Note that keeping current V LMC 0 (RR) and adopting fainter I LMC 0 (RC) would result in rather faint values of M LMC I (RC) ∈ (−0.13, −0.33), in potential disagreement with population models (see Girardi & Salaris 2000). This may suggest that either Udalski's (1998cUdalski's ( , 2000 dereddened clump magnitudes are more accurate or that dereddened V -magnitudes for RR Lyrae stars need revision.

Discussion
Using RR Lyrae stars and clump giants, I showed that the requirement of consistency between standard candles in different environments is a powerful tool in calibrating absolute magnitudes and obtaining distances. If the anomalous character of (V − I) 0 in Baade's Window is real (i.e., not caused by problems with photometry or misestimate of the coefficient of selective extinction), then the distance scale tends to be shorter. In particular, M V (RR) = 0.70 ± 0.05 at [Fe/H] = −1.6, and the distance modulus to the LMC spans the range from µ LM C = 18.24 ± 0.08 to 18.33 ± 0.07. If (V − I) 0 color of stars in Baade's Window is in error and should be standard, then the distance scale is longer. In particular, one can obtain M V (RR) = 0.59 ± 0.05 at [Fe/H] = −1.6 and the distance modulus from µ LM C = 18.35 ± 0.08 to 18.44 ± 0.07. It is important to notice that the reported distance modulus ranges do not change with the assumed value of the dereddened I-magnitudes of the LMC clump giants, I LMC 0 (RC).
Are there any additional constraints that would allow one to select the preferred value for RR Lyrae zero point β, M LMC I (RC), and µ LMC ? The fact that indirectly favors the intermediate distance scale (µ LMC ∼ 18.4) is its consistency with the results from classical Cepheids. The value of M V (RR) required for such solution is only 1.4 σ (combined) below the "kinematic" value of Popowski & Gould (1999) and 1.3 σ (combined) below the statistical parallax result given by Gould & Popowski (1998), leaving us without a decisive hint. The Twarog et al. (1999) study of two open Galactic clusters (NGC 2420 and NGC 2506) indicates rather bright red clumps. However, the relevance of this result to the LMC is uncertain and, more importantly, its precision is too low to provide significant information. The Beaulieu and Sackett (1998) study of clump morphology in the LMC suggests µ LM C ∼ 18.3, probably consistent with the entire (18.24, 18.44) range.
The only significant but ambiguous clue is provided by Udalski's (2000) spectroscopically-based investigation of the red clump in the solar neighborhood. One may entertain the following argument. If uncorrelated metallicity and age are the only population effects influencing M I (RC) in different environments (with age argued to have no effect in this case -Udalski 1998c) Table 1 would result in [Fe/H] LMC = −2.33! Such a low value is in violent disagreement with observations. Therefore, either uncorrelated metallicity and age are not the only population effects influencing M I (RC) (see Girardi & Salaris 2000 for a discussion) or Udalski (2000) results coupled with typical LMC metallicities lend strong support to the shorter distance scale. However, unless the selective extinction coefficient toward Baade's Window is unusual, very short distance scale comes at a price of anomalous (V − I) 0 bulge colors. Therefore, one is tempted to ask: "Is it normal that M I (RC) follows the local prescription and (V − I) 0 does not?".
In summary, with currently available photometry, it is possible to obtain the consistent RR Lyrae and clump giant distance scales that differ by as much as 0.2 magnitudes. Furthermore, even the presented distance scales may require some additional shift due to possible adjustments in α, M BW