Is there a formula for calculating the period/absolute magnitude relation for Mira variables in V or v (visua) band? Internet searches show J,H and K band versions but I don't find anything that could be used with AAVSO data.
the main problem will be to calculate the absolute magnitude for galactic Mira's since the distance is a problem (parallax limited) (not so for LMC and SMC AGBs and (not that much) for galactic bulge AGBs. A period vs color (for instance Ic-V) should do a better job (see Wood et al. IAU Symposium 191, 1999 MACHO observations of LMC red giants: Mira and semi-regular pulsators, and contact and semi-detached binaries, page 156, log P vs I0-1.38(V-I)0) )A similar plot can be made for galactic disk Mira's and SRBs. All the period/J, H and especially K-band relation formulae I know of are all based on LMC data, but the same formulae can be used for galactic LPVs. A good paper is Reprocessing the Hipparcos data of evolved stars III. Revised period-luminosity relationship for galactic long-period variable stars, A&A 403, 993-1002 (2003) Knapp et all. Here they give two formulae, one for galactic Miras and the other one for galactic SRs, but again for Mk data. The amplitude in K is much less than in V, so any use of formula(e) using (absolute) V should take into account the phase the star is in. I hope this can be of some help. Clear skies!
If there are references in the literature to a PL relation in V or v, I haven't been able to find them either.
This 2013 abstract summarizes recent results on an infrared PL: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8849856. Using VLBI and the "new" Hipparcos parallxes, the paper claims distances good to 10%. The first use of VLBI for that purpose is described in MNRAS, 2008, http://cdsads.u-strasbg.fr/abs/2008MNRAS.386..313W. Mira amplitudes are much larger in V than K, so a calibration in V, using the VLBI parallaxes and the recalibrated Hipparcos parallaxes might get a tighter Galactic PL. If there are visual and VLBI observations of the same Miras, it could be done - and the largest database of Mira visual light curves is in the AID! A nice data-mining project would be to identify which Miras are in common, followed up with a visual PL relation using AAVSO light curves for stars (if any) in common.
A general word of caution: pulsation characteristics are sensitive to metallicity, which varies amongst Galactic clusters, different galaxies, and within the "populations" of the Milky Way. There are Pop I and Pop II Cepheids, distinguished by metallicity and, hence, PL relations. The dependence of characteristics on metallicity is true of Miras.
Hi Mike et al.,
I looked at Period-Luminosity relations a lot as a graduate student, but not much since then, so some of my knowledge is a bit rusty. The main thing to remember is that there are two terms: period, which is easy to determine for a Mira using visual magnitudes, and luminosity, which is very difficult to get from visual estimates. Most of the energy output of a Mira is in the near-infrared, which is why K-band is often used. The visible spectrum is cut up with lots of molecular features and does not represent the changing radius of a pulsating star very well at all. As mentioned earlier, the other reasons for using K-band are that the star is brighter there (so can be seen at greater distances), and the light variation is smaller, so single-epoch observations yield something close to the luminosity of the star.
My feeble attempts at a V-band P/L for Miras failed miserably. It would be interesting to see what someone got today with a larger sample of Miras and more precise photometry, but my guess is random scatter, without other input data.
In another discussion in the LPV area a paper about Mira evolution models showed complex period changes which seemed to be reset at intervals by helium flashes. Nothing appeared to be mentioned about luminosity but that would be interesting in the present discussion.
My interest in some of these stars is period changes. R Hydrae, R Aquilae and R Centauri are thought to have just finished or are still in the throes of a Helium flash. Quite clearly the periods are shortening dramatically, but slowly, but is the luminosity? LX Cygni and BH Crucis, on the other hand, have undergone a lengthening of period in a tenth or so of that time, in both case roughly a 25% period increase in about 20 years. The values could be fitted by a mode change from the second to the first overtone. BH Crucis seems to have brightened and, accepting the B-V colour change as valid, become cooler and larger. Since it's a very red SC or CS star the visual amplitude is closer to the infrared, I imagine.
This is part of the background to which a P/L relationship must be fitted. But if the periods are slowly lengthening without - and this is difficult to establish since we're not around long enough - a luminosity increase, then an abrupt return to the starting period, it seems to me the possibility of such a relationship being found are remote. Then the physical structure of Miras is completely different from Cepheids where such a relationship has been found - although even here there are different groups.