Improved Baade-Wesselink surface-brightness relations

Recent, and older accurate, data on (limb-darkened) angular diameters is compiled for 221 stars, as well as BVRIJK[12][25] magnitudes for those objects, when available. Nine stars (all M-giants or supergiants) showing excess in the [12-25] colour are excluded in the analysis as this may indicate the present of dust influencing the optical and near-infrared colours as well. Based on this large sample, Baade-Wesselink surface-brightness (SB) relations are presented for dwarfs, giants, supergiants and dwarfs in the optical and near-infrared. M-giants are found to follow different SB-relations from non-M giants, in particular in V-(V-R). The preferred relation for non-M giants are compared to earlier relation by Fouque&Gieren (1997, based on 10 stars) and Nordgren et al. (2002, based on 57 stars). Increasing the sample size does not lead to a lower rms value. It is shown that the residuals do not correlate with metallicity at a significant level. The finally adopted observed angular diameters are compared to those predicted by Cohen et al. (1999) for 45 stars in common, and there is reasonable overall, to good agreement when \theta<6 mas. Finally, I comment on the common practice in the literature to average, and then fix, the zero point of the V-(V-K), V-(V-R) and K-(J-K) relations, and then re-derive the slopes. Such a common zero point at zero colour is not expected from model atmospheres for the (V-R) colour and depends on gravity. Relations derived in this way may be biased.


INTRODUCTION
The P L-relation of Cepheids is of fundamental importance in establishing the cosmic distance scale. Determining whether or not the slope of the Galactic relation is different from that for the LMC and/or SMC Cepheids relations is of prime importance, as this could imply a metallicity dependence of the slope of the relation, with important consequences for the application of the P L-relation to other galaxies. The apparent zero point and slope of the fundamental (FU) and first-overtone (FO) P L-relations in the LMC and SMC have now been well determined in V, I, W (the reddening free, so-called Wesenheit-index based on V and I), and J and K (Udalski et al. 1999, Groenewegen 2000, Nikolaev et al. 2004, Sandage et al. 2004).
The situation is less clear for our Galaxy. The accuracy of the HIPPARCOS parallax data was not high enough to determine slope and zero point. Instead, the data had to be analysed in a statistical way, such that for an assumed slope, a zero point could be derived (e.g., Feast & Catchpole 1997;Groenewegen & Oudmaijer 2000) There are basically two alternatives to obtain direct distance estimates to Galactic Cepheids, namely, using Cepheids in open clusters (e.g. Feast 1999, Tammann et al. 2003 where the distance to the cluster has been obtained in another way (basically main-sequence fitting), or, in combining linear diameter determinations (as determined from integration of the radial velocity curve and assuming a projection factor), with angular diameter determinations coming from direct measurements using an interferometer (e.g. Kervella et al. 2003) or coming from a surface-brightness (SB) relation and a reddening-corrected magnitude (e.g. Fouqué et al. 2003, Storm et al. 2004. For the latter method to work one has therefore to determine and calibrate accurate SB relations. There is a whole body of work on this subject (e.g., Barnes et al. 1978;Di Benedetto 1993 [DB93]), and more recently, with direct application to Cepheids, the works by Fouqué & Gieren (1997;herafter FG97), and Nordgren et al. (2002;hereafter Nord02). The latter calibration is used by Fouqué et al. (2003) to establish the most recent slopes and zero points of the Galactic P L-relation for FU Cepheids in BV IW JHK colours.
In addition, SB-relations are important in estimating angular diameters for planning interferometric observations (either to check if a potential science target would be resolved, or to check if a calibrator is indeed unresolved, for a given baseline), and for distance estimates in eclipsing binaries (Salaris & Groenewegen 2002).
The calibration by Nord02 is based on 57 giants observed with the Navy Prototype Optical Interferometer (NPOI) by this group (Nordgren et al. 1999;Nordgren et al. 2001). On the other hand there have been other recent papers presenting new diameter determinations (e.g., 69 stars in van Belle et al. 1999), and there exist older data of high quality; recently, Richichi & Percheron (2002) Nordgren et al. (2001) plotted against those from Mozurkewich et al. (2003) for the stars in common. The line is a least-squares fit to the 15 data points plotted: θ N2001 = (1.002 ± 0.007) θ M2003 + (0.000 ± 0.031) (rms = 0.039). Table 2. Objects with three or more independent angular diamter estimates.

ID
θ ± σ θ θ ± σ θ θ ± σ θ θ ± σ θ (mas) (mas) (mas) (mas) conveniently presented a catalog of high angular resolution measurements. The aim of the present paper is to present updated SB relations based on the largest set of accurate angular diameter determinations. Section 2 presents the angular diameter data and the search for apparent magnitudes. Section 3 briefly recalls the relevant equations, and Section 4 presents the results. The discussion in Section 5 ends this paper.

THE DATA
The principal sources of limb-darkened angular diameters are the recent papers by Mozurkewich et al. (2003), Nordgren et al. (1999Nordgren et al. ( , 2001, van Belle et al. (1999), Kervella et al. (2003Kervella et al. ( , 2004 and Wittkowski et al. (2004). From the compilation by Richichi & Percheron (2002), those stars were selected which have a limbdarkened angular diameter determined, and with a relative accuracy of better than 3%. This selection is based on the typical accuracy that can be achieved with modern instrumentation and was selected to weed out most of the less accurate, older, data in this compilation. The stars already in the other references were removed, as well as the carbon-and S-stars. In total this added another 27 objects. The final list contains 221 (different) objects, listed in Table 1. In case of multiple angular diameter determinations the one with the smallest relative error on the limb-darkened angular diameter was retained.
For fifteen objects there are three or four independent determinations available and its interesting to compare them as e.g. slightly different approaches are used in the literature to proceed from uniform-disk to limb-darkened disk angular parameters. Table 2 lists those stars and the measurements, and Figure 1 plots the values from Nordgren et al. (2001) against those in Mozurkewich et al. (2003). The agreement is extremely good, and the rms in the fit of 0.039 indicates that the quoted errorbars in the determinations are realistic.
Apart from the angular diameters, one needs apparent magnitudes and reddening to apply the SB method. Most of the photometric data come from consulting the SIMBAD database. In particular, IRAS 12 and 25 µm fluxes, and BV RIJK magnitudes were collected from the entries marked (in order of preference), "JP11", "UBV", "IRC". In addition, the galactic coordinates, spectral type, and the HIPPARCOS parallax was retrieved, as well as the most recent listed value for the metallicity, available under the item marked "Fe H". To collect more photometric data, the Gezari et al. (1999) catalog was checked for infrared data, and the SIMBAD Table 1. Identification (HR or IRC identifier, unless otherwise listed in the remarks), adopted angular diameter and error, adopted visual reddening and error, BV RIJK [12][25] photometry, reference to the angular diameter and photometry (when not taken from SIMBAD as discussed in the text), and remarks, of the sample studied. An entry with −9.99 means that this magnitude is not available.   (1996) based list of references was checked for papers that, based on the title alone, might contain additional photometry. For 43 stars only a K-mag from the IRC-survey is available, and it was investigated if there is a systematic difference between Johnson K and IRC K. Based on 66 objects (minus 3 outliers) a difference K(Johnson -IRC) = 0.017 ± 0.045 was established, and therefore no correction to the IRC magnitudes was applied. It may appear attractive to include 2MASS photometry for all stars-for reasons of uniformityor at least for those stars without infrared photometry sofar. However, the stars in the sample are so bright that no 2MASS photometry could be obtained in the standard way. Figure 2 shows for 119 stars with JK photometry on the Johnson system and 2MASS photometry with a quoted errorbar in both filters of less than 0.3 mag the correlation between (J − K) Johnson and (J − K)2MASS and K Johnson − K2MASS and (J − K)2MASS. The rms in the fits are about 0.1 mag, and so applying these relations would cause un-wanted scatter, and therefore 2MASS data has not been used. It should be noted that some "natural" scatter is present in the adopted photometry because e.g. of non-simultaneous photometry or low level variability.
The finally adopted magnitudes are listed in Table 1, and are predominantly on the Johnson system. The IRAS flux densities were converted to magnitudes assuming zero points of 41.0 and 9.49 Jy at 12 and 25 µm respectively.
The reddening was estimated using the reddening model by Arenou et al. (1992), which returns the reddening estimate (with error) based on galactic coordinates and distance. The distance used was simply based on the HIPPARCOS parallax, and when no parallax was available a distance of 1 kpc was assigned. As discussed later, none of these assumptions is critical to the final results of this paper. The adopted values for AV and its error are listed in Table 1. The relative reddening A λ /AV are adopted to be 1.33, 0.80, 0.49, In what follows we will the use the following terminology: giants are objects which have "III" in their spectral type listed in Table 1 (i.e. includes stars with spectral type II-III), dwarfs are objects of luminosity class V, sub-giants are objects of luminosity class IV, supergiants are objects of luminosity class I, and luminous giants are objects of luminosity class II. Three objects have no luminosity class assigned in SIMBAD and for those the absolute V -magnitude and (B − V )0 were determined from the parallax and reddening, and compared to the tables of Straižys & Kuriliene (1981) and Schmidt-Kaler (1982). Based on this, HR 6208 (MV = 0.3 ± 0.2) was assigned luminosity class III, and HR 8555 (MV = −1.5 ± 0.4) and IRC +40 337 (MV = −1.6 ± 0.8) were assigned luminosity class II.

BASIC RELATIONS
A surface-brightness relation can be defined as follows (see van Belle 1999): where θ is the (limb-darkened) angular diameter (in mas), and m1 a de-reddened magnitude (for example, V ). The logarithm of this quantity is plotted against a de-reddened colour (for example, and a linear fit is performed. An equivalent form is to use the quantity (see, e.g., Nord02) and to plot this against a colour; The former mathematical formulation will be used in this paper, but it is trivial to show that, In the analysis below, the error on θ0 includes the error on θ, an arbitrary but representative error of 0.01 mag in the photometry, and the error on the reddening.

RESULTS
The assumption in the above relations is that, after correction for interstellar reddening, the colours represent the photospheric colours.
As a large fraction of the stars among the sample of 221 are M giants (59) or supergiants (7) one might need to consider the influence of mass loss on the colours. Following the literature, the following SB relations are considered: V versus (V −R), V versus (V −K), and K versus (J −K) (e.g. Fouqué & Gieren 1997). The results of the fitting are listed in Table 3 and shown in Figures 4-12. The table lists the values of the fitting coefficients a and b, the number of stars considered, the rms in the fit, the colour range over which the fit was performed, the luminosity class, and some additional remarks. In the bottom panel of all figures the residual is plotted against metallicity, and the fit is explicitly given when significant. The possible significance of this will be discussed in Section 5.
The fit was performed using a least-squares algorithm. Outliers were identified and removed, and the fit repeated. Outliers were again removed and the final fit made. An object was considered an outlier if the distance between a data point and the fitted line was more than 5 times either the error bar in the data point, or the rms in the fit. Table 3 also shows some results when the σclipping is more stringent. The rms in the fit are obviously reduced but not by much, and the coefficients change within their error bars.  Table 3 also includes the result when not considering the Mgiants. In that case the colour range over which the relation is applicable becomes smaller, but the relations might be more appropriate for Cepheids for which these relations are often used. The number of stars in the sample is reduced and hence the formal error on the coefficients becomes slightly larger. The rms in the fits are not appreciably reduced. Considering only the M-giants results in a relation which is significantly different from the relation for all giants, and all non-M giants, as suggested already by DB93.
To investigate the influence of reddening a set of 1000 Monte Carlo simulations was performed where each time the reddening was replaced by a value drawn from a Gaussian distribution with mean the adopted reddening and sigma the error therein. The 23rd and 977th ordered value correspond to ±2σ, and in the case of the giants (first entry in Table 3), the value and the error on a and b are (0.2368 ± 0.0005, 0.6080 ± 0.0012), (0.7279 ± 0.0011, 0.6076 ± 0.0010) and (0.1607 ± 0.0038, 0.5905 ± 0.0030), for the case V versus (V − K), V versus (V − R) and K versus (J − K), respectively. The conclusion is that, formally, the influence of reddening is smallest in the case V versus (V − K). However, in all three cases the error in the reddening is insignificant compared to the fit error.

DISCUSSION
Surface brightness relations have been derived for a large number of stars making use of the large datasets on angular diameter measurements for "normal" stars that have become available recently. In the present section the results will be discussed, and compared to earlier work. For convenience, SB relations reported in the literature have been compiled in Table 4 following the format of Table 3 (usually a ′ and b ′ are quoted, and they have been converted to a and b by me. The rms is the one quoted for the original fit-which therefore almost always refer to Eq. (4)-, and its value should be  multiplied by a factor of 2 to compare to the rms values listed in Table 3 !) Recently, Kervella et al. (2004) presented SB-relations for stars of luminosity class IV + V of the type m1 versus (m1−m2) for all possible combinations for magnitudes U BV RIJHKL. The agreement is perfect in the case V − (V − K) but this should not be surprising as essentially the same angular diameter and photometric data has been used. For the other 2 relations the fits not in good agreement. In those two cases the clipping at 4σ has removed a substantial number of objects in the present paper, while Kervella Table 3. SB relations derived in this work. For the giants the solutions marked by a • are the preferred ones. For the supergiants a comparison can be made with FG97. For the V − (V − K) and V − (V − R) relation the sample considered here is almost double that of FG97, and extends to bluer colours. The error on the coefficients is correspondingly smaller, yet the rms is in fact slightly larger. For typical colours (V − K)0 = 2.0, (V − R)0 = 0.5, and (J − K)0 = 0.5 the relation in the present paper predicts log θ0 of 1.093, 0.913 and 0.668, respectively, while FG97 predict 1.089, 0.947 and 0.666, respectively. There are some differences but not systematically so it appears. The relations in the present paper should be preferred because they are based on a larger number of stars.
Probably of most interest is the relation for giants. This is also the class of stars for which most data is available. A first important conclusion is that the relation for the M-giants is different from that for all giants and for all non-M giants. This was first hinted at by DB93 based on few stars only, but is confirmed here based on a much larger sample. Like him it is found that a is smaller and b larger for M-giants than for the non-M giants, in particular for the V − (V − R) and K − (J − K) relations at a level of 3-5σ. This makes a comparison with FG97 and Nord02 difficult as they did not exclude M-giants in deriving their SB-relations. For typical colours (V − K)0 = 2.5, (V − R)0 = 0.9, and (J − K)0 = 0.7 the preferred relation for non-M giants in the present paper predicts log θ0 of 1.197,1.258 and 0.698,respectively,while FG97 (Nord02) predict for (all) giants 1.201 (1.188), 1.272 (1.248), 0.701 (0.690). For all three relations the sizes are slightly smaller than predicted by FG97-which is consistent with the fact that it is found in the present paper that the M-giants are bigger than the non-M giants-but smaller than the values listed in Nord02 which includes M-giants.
As was noted earlier in the literature, including more stars does not lead to a smaller rms. Considering giants of all types, the rms in, for example, the V − (V − K) relation is 0.016 (based on 10 stars in FG97), 0.022 (based on 57 stars in Nord02) and 0.027 (based on 122 stars in the present paper). It is shown that the SB relations do not significantly depend on metallicity, and hence this scatter is not due to that parameter. Specifically for the giants, only  the residuals in the V − (V − R) relation correlate at the 1σ level with metallicity (see caption of Figure 5). Cohen et al. (1999) predict angular diameters for more than 400 giants in the spectral range G9.5 to M0 by fitting model atmospheres to absolute flux-calibrated broad-band photometry. Already in their paper they compared the predicted values to observed values, for about 20 stars. Figure 13 shows the same comparison but now for a sample twice in size. A least-square fit was made and outliers at the 4σ level removed (HR 2012, 4546). A new fit was made, and one additional outlier (HR 7942) was removed. The final fit is: θ observed = (1.049 ± 0.010) θ model + (−0.166 ± 0.048),  with an rms of 0.130. The slope is in the same sense as found by Cohen et al. (1.013 ± 0.008), i.e. slightly above unity, and found to be slightly steeper now, while the zero point is now significantly below zero, while the value in Cohen et al. was consistent with zero (0.035 ± 0.073). No immediate explanation is offered but it is simply noticed that the rms in the fit is larger than expected based on the assigned errorbars. As independent observations seem to be in very good agreement (see Figure 1) any possible problem seems more likely to be related to the models rather than the ob-  servations. In addition, any possible problem appears to be for the larger objects. Selecting only stars smaller than a certain cut-of and monitoring the significance of the derived slope and zero point it is found that for θ < 6 mas the expected 1-to-1 relation is found θ observed = (1.009±0.019) θ model +(−0.028±0.066), although the rms is only reduced to 0.117 mas.
A final point that has to be addressed is the practice in the literature (FG97 and Nord02) to impose a common zero point of SB relations. In practice, the same three relations used here (i.e. Table 5. Colours of normal solar metallicity stars (from Pietrinferni et al. 2004). V − (V − K), V − (V − R) and K − (J − K)) are first derived independently. Then the zero point is averaged. Finally new slopes are derived for the three relations keeping the zero point fixed to this averaged value. The argument given is along the line that "For a star of spectral type A0, where (V −R)0 = (V −K)0 = (J −K)0 = 0, each of the relations should yield a common surface brightness" (Nord02). Life is not that simple. Table 5 lists the (theoretical) colours of stars at various effective temperatures and gravities very near the colour (V − K) = 0 (from Pietrinferni et al. 2004). It can be remarked that indeed (J − K) is in all cases also very near zero, but (V − R) is near zero only for main-sequence stars but not for lower gravities. Imposing a common zero point based on simple averaging individual zero points is therefore not allowed for giants or supergiants. The value derived in such a way is biased (and this will also influence the then newly calculated slopes) if the common zero point is based on the zero point from the V −(V −R) relation, like is the case in FG97 and Nord02.