Planetary nebula luminosity functions

The CASt dataset

Astronomical background

The nomenclature "planetary nebula" is historical and is misleading: it has nothing to do with planets.  Rather, planetary nebulae (PNe) were discovered by 19th century astronomers as luminous structures, often circular or ring-shaped, in the sky with a very hot central star.  In the 20th century, they were understood as a phase in the evolution of intermediate-mass stars like our Sun: after the main sequence star depletes hydrogen in its core over billions of years, it becomes a red giant star with complicated stages of hydrogen/helium/carbon fusion.  When these stages terminate, the outer layers of the star are ejected into a PN, revealing the hot core which cools to become a white dwarf star.  PNe are thus studied in detail to understand the death of stars.

The PN phase is very short-lived, so that only ~102 of the ~1011 stars in a major galaxy exhibit PNe at a given moment.  For complex reasons, the collective distribution of PN luminosities appears to be a universal property of galaxies.   This distribution is called the planetary nebula luminosity function (PNLF) and is one of the principal tools used to estimate distances to nearby galaxies.  Distances are directly related to the offset It is thus a step in the "cosmic distance ladder" which determines the scale, expansion and age of the Universe. 

Dataset

We give here simple univariate datasets of PN brightnesses in magnitudes for five nearby galaxies.  The magnitudes are measured in [O III]5007, a "forbidden" emission line of twice-ionized oxygen.  Messier 31 = Andromeda Galaxy (238 PNe), the closest large galaxy and distance scale calibrator); Messier 81 (89 objects), a large spiral just outside our Local Group of galaxies;  NGC 3379 (45 objects), a nearby elliptical galaxy;  NGC 4494 (101 objects) and NGC 4382 (59 objcts) in the Virgo Cluster, the nearest rich cluster of galaxies.  Magnitudes are an inverted logrithmic unit of brightness.  Due to observational limitations at the telescope, only the brighter PNe in a given galaxy can be detected.  Each observation has a "completeness" limit which truncates the PNLF (which must be estimated), and usually includes some PNe which are fainter than this limit.

Original references for these datasets and PNLF studies are:
    M31 (bulge):  Ciardullo et al. 1989, Ap.J., 339, 53   +
                        Ciardullo et al. 2002, Ap.J., 577, 31
    M81 (bulge):  Jacoby et al. 1989, Ap.J., 344, 704
    NGC 3379:     Ciardullo, Jacoby, & Ford 1989, Ap.J. 344, 715
    NGC 4494:     Jacoby, Ciardullo, & Harris 1996, Ap.J., 462, 1
    NGC 4382:     Jacoby, Ciardullo, & Ford 1990, Ap.J., 356, 332

The first figure below (from Ciardullo et al. 1989) shows the PNLF of M 31 using the complete (untruncated) sample.  The data have been grouped into 0.2 mag wide bins.  The segmented line is an independently derived model (not based on the data) from an astrophysical calculation of PN evolution.

The figure below (from Ciardullo et al. 2002) shows several examples of PNLFs (in grouped magnitudes), the "universal" PNLF shape, and the offsets between galaxies due to different distances.  Only the filled circles lie above the magnitude truncation limit and only these are used to estimate galaxy distances.

Statistical exercises

• Using nonparametric and parametric techniques, test the hypothesis that the the individual PNLFs are drawn from a single universal distribution.  That is, the PNLFs have the same shape though different magnitude offsets and truncations.  In particular, is the PNLF of the elliptical galaxy at low metallicity consistent with the PNLF of the higher-metallicity spiral galaxies?  Use only the PNe brighter than (i.e. smaller magnitude values) than the truncation limit. 
  
• Using semi-parametric and parametric techniques, estimate the magnitude offset between M 31 and the other galaxies.  Estimate the confidence intervals of each offset.  Compare results to published distances to these galaxies. 
• Determine a parametric form for the PNLF with best-fit parameters and confidence intervals.  Use this to quantitatively estimate the magnitude truncation limit for each galaxy.
• Using semi-parametric or parametric techniques, estimate the offsets between the galaxy PNLFs in number (rather than magnitude), which gives insight into the total stellar populations of the galaxies.