4. The PL relation and LMC distance modulus
The PL relation of Miras can be used to determine the distance modulus to LMC (van Leeuwen et al. 1997). Here, we calculate the slopes and zero-points of the mean PL relation for Sample 1. Some care is needed since gaussian parallax errors propagate into non-gaussian errors in distances or absolute magnitudes and a bias may result (Smith & Eichhorn 1996). To avoid this, one can write:
where and The estimated true annual parallax of the star, in mas, is taken from Table 1. The mean (variations) dereddened apparent magnitude is , the mean (stat.) absolute magnitude is that a 1 day-period star would have, and s is the slope of the PL relation. A non-linear fit of Eq. (1) was computed from the E04HFF routine of the NAG software to minimize:
where the are the data weights. We have used successively (unweighted) and where K is a constant, which is the usual choice. Having derived a and b, the corresponding s -value is replaced in:
and the absence of correlation between the computed right hand side and is checked for. The best results were obtained from weighted data and the corresponding values of s and are given in Table 2. Fits in Fig. 1b (C-LPVs from references in Sect. 3.1) and from Reid et al. (1995, O- and C-LPVs) are also given which show nearly identical s -slopes for the Galaxy and LMC. It is thus advisable to deduce the LMC distance modulus by difference of the intercepts. We obtain . No correction was applied for metallicity as suggested by current evidence from various stellar systems (Feast et al. 1996). The modulus quoted in Table 2 would otherwise be a lower limit. The slopes of the upper limit (ul in Fig. 1) were similarly calculated (Figs. 1a: 22 stars including 14 with appreciable weight and 1b: 13 stars). Values of -4.85 in the Galaxy and -4.72 in the LMC were found. No estimate of the distance modulus is attempted here.
Table 2. The PL-relation of galactic and LMC carbon LPVs (s is the slope, the intercept and n the number of stars). Eq. (3): slope vs. , intercept and correlation coefficient.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998