8.2 The Planetary-Nebula Luminosity Function

In his study of Magellanic Cloud planetaries, Jacoby (1980) showed that the [O III] 5007 luminosity function for faint planetaries is well represented by an exponential law formed by combining a uniformly expanding shell with a non-evolving central star (Henize and Westerlund 1963). However, as the CCD observations of PN in M31 (Ciardullo et al. 1989), M81 (Jacoby et al. 1989), the Leo I Group (Ciardullo et al. 1989), the NGC 1023 Group (Ciardullo et al. 1991), the Magellanic Clouds (Jacoby et al. 1990), and Virgo (Jacoby et al. 1990) have shown, the bright end of this function is truncated rather dramatically, in a manner similar to that predicted by the theoretical models of Jacoby (1989). Ciardullo et al. (1989) therefore modified the Henize and Westerlund law by adding a cutoff exponential, proposing that

(18)

where the magnitude, M, is related to the 5007 flux of a planetary by

(19)

and M* is the absolute magnitude for the most luminous planetary.

To set the zero point of the function, Ciardullo et al. (1989) used 104 unobscured PN in the bulge of M31. By adopting the infrared Cepheid distance of 710 kpc (Welch et al. 1986), and a total foreground extinction of A₅₀₀₇ = 0.39, Ciardullo et al. found the best fit value for M* to be M* = -4.48 with a 1 internal error of +0.036, -0.046. Because these 104 PN serve as the sole absolute calibrator for PNLF distances, this internal error, as well as the external uncertainty in M31's true distance and reddening, propagates directly into all subsequent PNLF measurements. Future observations may reduce the internal error by enlarging the sample of PN used in the calibration, but because PN are secondary standard candles, PNLF distances will always be subject to uncertainties in the Local Group distance scale. In fact, recent estimates for M31's distance (770 kpc; Freedman and Madore 1990) and reduced extinction (A₅₀₀₇ = 0.28; Burstein and Heiles 1984) suggest a cutoff that is 0.06 mag more luminous.

Since the Ciardullo et al. PNLF is not a power law, and, in fact, shows a rapid decline over the brightest magnitude of the luminosity function, a fit of the observed PNLF to equation (18) yields a distance directly, provided, of course, that the function is invariant. Fortunately, there is good evidence to show that this function, if not invariant, is at least insensitive to galaxy color, metallicity, or Hubble type. Well defined PNLFs currently exist for 17 giant spiral, elliptical, and irregular galaxies. In none of these objects is there any evidence for a change in the shape of the PNLF cutoff (cf. Figure 15). More importantly, the zero point of this function also appears to be constant. In the Leo I Group, Ciardullo et al. (1989) found that the PNLF distances for three elliptical and S0 galaxies, NGC 3377, 3379, and 3384, all fell within 0.5 Mpc, or 5%, despite the fact that the galaxies differ by 0.8 mag in absolute luminosity, 0.3 mag in (U - V) and 0.1 in [Fe/H]. Jacoby et al. (1990) derived a similar result for six early-type galaxies in the Virgo Cluster core: although the galaxies span a range of 0.19 in [Fe/H], 0.22 mag in (U - V), and 2.2 mag in (m₁₅₅₀ - V), the 1 dispersion in their computed distances was again only 5% (0.8 Mpc). Perhaps the strongest internal confirmation of the insensitivity of the PNLF to stellar population, however, comes from the agreement between the PNLFs of NGC 1023, a gas-rich S0 galaxy, and fellow group member NGC 891, the edge-on Milky Way lookalike. Ciardullo et al. (1991) measured identical distances to these two galaxies, using as their data a complete sample of PN in NGC 1023's disk and NGC 891's halo. Since the halo stars of NGC 891 are undoubtably much older than the disk objects of NGC 1023 (NGC 1023 formed stars ~ 3 Gyrs ago [Gregg 1989]), this result strongly supports the contention that a galaxy's star formation history does not play an important role in determining the PNLF cutoff. Moreover, since the total number of PN found in NGC 891's halo is a factor of three less than that discovered in NGC 1023, the similarity in the derived distances to these two galaxies is additional evidence that the PNLF cutoff is indeed an exponential, rather than a power law as proposed by Bottinelli et al. (1991).

Figure 15. The observed planetary-nebula luminosity functions for M31, M81, three galaxies in the Leo I group, and six galaxies in the Virgo cluster. Aside from small differences caused by the photometric errors associated with each observation, the shapes of the four functions are identical, demonstrating the form of the PNLF does not depend on Hubble type. Open circles display points which were not used in the solution. The solid lines show the best empirical luminosity functions.

External comparisons have also proved that PNLF distances do not depend strongly on the properties of the underlying stellar population. PNLF surveys in the bulge of the Sb galaxy M81 and in the Large Magellanic Cloud have resulted in derived distances to these galaxies (Jacoby et al. 1989; Jacoby et al. 1990) that are statistically indistinguishable from that found from the I-band and infrared observations of Cepheids (cf. Freedman and Madore 1988; Welch et al. 1987; Feast and Walker 1987). Moreover, Pottasch's (1990) PNLF based estimate of 8.1 kpc for the distance to the Galactic Center is completely consistent with that determined by other methods (Reid 1989). Finally, it should be noted that the PNLF distances to the Leo I, NGC 1023, and Virgo clusters are in excellent agreement with those estimated from the Tully-Fisher method (Pierce and Tully 1989; Aaronson and Mould 1983), and a comparison between the PNLF and surface brightness fluctuation distances for 14 galaxies measured by both methods yields a dispersion of only 0.19 mag, a number consistent with the internal error estimates of the two methods (see Sec. 11).