6.4
The dashed line in Figure (6.5) represents the observed absolute magnitude (M𝑟) distribution of
SampleMetal, showing a considerable number of stars with absolute magnitudes between +8 and +12 mag.
86
A transient spiral arm appears and disappears over time, as contrasted with a stationary spiral arm which is long lived and persistent. Stationary spiral arms do not affect the stellar motion in the vertical direction significantly (Nordstrӧm 2008).
87
Recently, West et al. in the Cool Stars 18, 2014 applied a new photometric metallicity to a very large sample of M dwarfs and the decrease in metallicity as a function of Galactic height was nicely represented.
87 Figure (6.5) The observed absolute magnitude distribution of stars in SampleMetal (the dashed histogram) and the volume-corrected distribution (the solid histogram).
Similar to the metallicity distribution, we determined the absolute magnitude distribution of each Galactic height bin and then added them together, leading to the volume-corrected distribution of the entire sample. The fractional volume-corrected absolute magnitude distribution is illustrated as a solid line in Figure 6.5. In a magnitude-limited sample88, the faint stars are more detected in closer distances. By the volume correction described above, the contributions of stars in smaller heights were increased while those of larger heights were decreased. As a result, the volume- corrected absolute magnitude distribution is fainter than the uncorrected one.
To trace the relation between metallicity and absolute magnitude, we determine the volume- corrected absolute magnitude distributions of stars in SampleMetal with different metallicity ranges, as shown in Figure (6.6). Our results indicate that the distributions of metal-poor M dwarfs have higher fractions of brighter stars than those of metal-rich M dwarfs. However, we cannot readily conclude that metal-poor dwarfs are more luminous than metal-rich ones as the difference between the luminosity distributions of different metallicity intervals could be, totally
88
In magnitude-limited samples, stars below a certain apparent brightness are not included.
88 or partially, a consequence of the existing bias in our magnitude-limited sample. As already stated, the metallicity distribution of M dwarfs tends to be more metal-poor as height increases. This means that the metal-poor M dwarfs are more likely to be found at higher Galactic heights and, on average, have larger distances than metal-rich dwarfs. In a magnitude limited sample, the detected lower-metallicity M dwarfs are, on average, more luminous than higher-metallicity one. To find a more accurate statistical relation between absolute magnitude and metallicity, the effect of this bias must be removed.
Figure (6.6) The volume-corrected absolute magnitude distributions for 8 subsamples of different
Some studies favour a decrease in luminosity as a function of metallicity. For example, Laughlin (2000) argued that since metal-poor stars are brighter than their metal-rich counterparts of the same mass, there was no observational bias against metal-poor stars in his study. He also pointed out that doubling the metallicity of a solar-mass star from zero to +0.3 dex leads to a 320 K decrease in effective temperature and a factor of 1.3 decrease in luminosity. Using theoretical models of low-mass stars, Cassisi (2011) showed that the HR locus became cooler and fainter with increasing heavy element abundances. We applied the isochrones for low-mass stars of age 5 Gyr (the mean age of Galactic disk) and three different assumed metallicities (0.0, -0.5, and -1.0 dex) taken from Baraffe et al. (1998) and then plotted the luminosity versus mass for these
89 isochrones, as shown in Figure (6.7). It is clear that at a given mass, the lower-metallicity stars are more luminous than higher-metallicity ones.
To the contrary, Woolf & West 2012 took a different approach and argued that since low- metallicity main sequence stars (subdwarfs) are less luminous than higher-metallicity stars of the same temperature or spectral type, higher metallicity stars should be overrepresented in magnitude-limited samples. In other words, metal-poor M dwarfs must be, on average, closer than the metal-rich ones in such samples. Figure (6.8) represents the luminosity versus temperature of the isochrones described above, showing an increase in luminosity as metallicity rises for a given temperature. Using a sample of stars from Woolf et al. (2009) for which parallax data were available, Woolf & West (2012) derived the luminosity variation with metallicity for M dwarfs in the temperature range 3500 ≤ Teff ≤ 4000 K. Based on this variation, they calculated a volume correction factor for different values of [Fe/H] (Table 1 in their paper). Another example which supports this view is the work of Schlesinger et al. (2012) on a sample of G and K dwarfs. In this study, it was noted metal-rich stars are brighter than metal-poor ones for a given g-r colour. Therefore, the volume coverage of the sample varies with respect to metallicity for the same magnitude range. Schlesinger et al. specified distance limits for their sample to ensure that there was no bias due to the metallicity-dependent volume coverage.
Figure (6.7) The luminosity versus mass for the isochrones of age 5 Gyr and three different metallicities. The metallicities are colour-coded and the masses are symbol-coded.
90 Figure (6.8) The luminosity versus temperature for the isochrones of age 5 Gyr and three different metallicities. The metallicities are colour-coded and the masses are symbol-coded.
Considering the two opposite views described above, in order to make an appropriate volume correction to a magnitude-limited sample like ours, more investigations on the relation between absolute magnitude and metallicity are required.