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      <td style="vertical-align: top;">&nbsp; </td>
      <td>
      <h2 style="text-align: center;">Planetary nebula luminosity
functions<br>
      </h2>
      <h3>The CASt dataset</h3>
      <a href="plan_neb.dat" target="_blank">plan_neb.dat</a><br>
      <p> <span style="font-weight: bold;">Astronomical background </span></p>
      <p>The nomenclature "planetary nebula" is historical and is
misleading: it has nothing to do with planets.&nbsp; 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.&nbsp; 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.&nbsp; 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.&nbsp; PNe are thus studied in
detail to understand the death of stars.<br>
      </p>
      <p>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.&nbsp; For
complex reasons, the collective distribution of PN luminosities appears
to be a universal property of galaxies. &nbsp; This distribution is
called the planetary nebula luminosity function (PNLF) and is one of
the principal tools used to estimate distances to nearby
galaxies.&nbsp; 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.&nbsp; <br>
      </p>
      <p><span style="font-weight: bold;">Dataset</span></p>
      <p>We give here simple univariate datasets of PN
brightnesses in magnitudes for five nearby galaxies.&nbsp; The
magnitudes are measured in [O III]5007, a "forbidden" emission line of
twice-ionized oxygen.&nbsp; 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;&nbsp; NGC 3379
(45 objects),
a nearby elliptical galaxy;&nbsp; NGC 4494 (101 objects) and NGC 4382
(59 objcts) in the Virgo
Cluster, the nearest rich cluster of galaxies.&nbsp; Magnitudes are an
inverted logrithmic unit of brightness.&nbsp; Due to observational
limitations at the telescope, only the brighter PNe in a given galaxy
can be detected.&nbsp; 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. <br>
      </p>
      <p>Original references for these datasets and PNLF studies are:<br>
&nbsp;&nbsp;&nbsp; M31 (bulge):&nbsp; Ciardullo et al. 1989, Ap.J.,
339, 53&nbsp;&nbsp; + <br>
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; Ciardullo et
al. 2002, Ap.J., 577, 31<br>
&nbsp;&nbsp;&nbsp; M81 (bulge):&nbsp; Jacoby et al. 1989, Ap.J., 344,
704<br>
&nbsp;&nbsp;&nbsp; NGC 3379:&nbsp;&nbsp;&nbsp;&nbsp; Ciardullo, Jacoby,
&amp; Ford 1989, Ap.J. 344, 715<br>
&nbsp;&nbsp;&nbsp; NGC 4494:&nbsp;&nbsp;&nbsp;&nbsp; Jacoby, Ciardullo,
&amp; Harris 1996, Ap.J., 462, 1<br>
&nbsp;&nbsp;&nbsp; <small>NGC 4382:&nbsp;&nbsp;&nbsp;&nbsp; Jacoby,
Ciardullo, &amp; Ford 1990, Ap.J., 356, 332<br>
      </small></p>
      <p><small>The first figure below (from Ciardullo et al. 1989)
shows the PNLF of M 31 using the complete (untruncated) sample.&nbsp;
The data have been grouped into 0.2 mag wide bins.&nbsp; The segmented
line is an independently derived model (not based on the data) from an
astrophysical calculation of PN evolution.<br>
      </small></p>
      <div style="text-align: center;"><img alt="M 31 PNLF"
 src="plan_neb_fig1.gif" style="width: 507px; height: 402px;"><br>
      </div>
      <p><small><br>
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.&nbsp; Only the
filled circles lie above the magnitude truncation limit and only these
are used to estimate galaxy distances.<br>
      </small></p>
      <div style="text-align: center;"><img
 alt="Planetary nebula luminosity functions" src="plan_neb_fig.gif"
 style="width: 567px; height: 704px;"><br>
      </div>
      <p><small> </small></p>
      <p><span style="font-weight: bold;">Statistical exercises<br>
      </span> </p>
      <ul>
      </ul>
      <ul>
        <li>Using nonparametric and parametric techniques, test the
hypothesis that the the individual PNLFs are drawn from a single
universal distribution.&nbsp; That is, the PNLFs have the same shape
though different magnitude offsets and truncations.&nbsp; In
particular, is the PNLF of the elliptical galaxy at low metallicity
consistent with the PNLF of the higher-metallicity spiral
galaxies?&nbsp; Use only the PNe brighter than (i.e. smaller magnitude
values) than the truncation limit.&nbsp; <br>
        </li>
        <li>Using semi-parametric and parametric techniques, estimate
the magnitude offset between M 31 and the other galaxies.&nbsp;
Estimate the confidence intervals of each offset.&nbsp; Compare results
to published distances to these galaxies.&nbsp;</li>
        <li>Determine a parametric form for the PNLF with best-fit
parameters and confidence intervals.&nbsp; Use this to quantitatively
estimate the magnitude truncation limit for each galaxy.</li>
        <li>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.<br>
        </li>
      </ul>
      <br>
      <div style="text-align: center;"><small><span
 style="font-style: italic;">These datasets were kindly prepared by
Prof. Robin Ciardullo, Penn State University</span><br>
      </small></div>
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