AGB variables and the Mira period–luminosity relation

Published data for large-amplitude asymptotic giant branch variables in the Large Magellanic Cloud (LMC) are re-analysed to establish the constants for an infrared ( K ) period–luminosity relation of the form   M _K =ρlog  P ? 2.38]+δ  . A slope of  ρ=?3.51 ± 0.20  and a zero-point of  δ=?7.15 ± 0.06  are found for oxygen-rich Miras (if a distance modulus of 18.39 ± 0.05 is used for the LMC). Assuming this slope is applicable to Galactic Miras we discuss the zero-point for these stars using the revised Hipparcos parallaxes together with published very long baseline interferometry (VLBI) parallaxes for OH masers and Miras in globular clusters. These result in a mean zero-point of  δ=?7.25 ± 0.07  for O-rich Galactic Miras. The zero-point for Miras in the Galactic bulge is not significantly different from this value.
Carbon-rich stars are also discussed and provide results that are consistent with the above numbers, but with higher uncertainties. Within the uncertainties there is no evidence for a significant difference between the period–luminosity relation zero-points for systems with different metallicity.