Toward Precision Cosmology with Improved Planetary Nebula Luminosity Function Distances Using VLT-MUSE. II. A Test Sample from Archival Data

The shape of the PNLF is not universal: in old stellar populations, the slope at faint magnitudes changes from galaxy to galaxy (e.g., Longobardi et al. 2013; Hartke et al. 2017, 2020; Bhattacharya et al. 2021), while in star-forming systems, the PNLF is known to contain multiple inflection points (Jacoby & De Marco 2002; Ciardullo 2010; Reid & Parker 2010). However, at the extreme bright end of the function, the PNLF's behavior is remarkably invariant and consistent with an abrupt truncation of an underlying power law. Moreover, measurements of roughly a dozen galaxies within ∼10 Mpc show that for systems more metal-rich than the LMC, the absolute magnitude where this truncation takes place is extraordinarily insensitive to stellar population (e.g., Ciardullo 2013). Thus, the traditional method of deriving a PNLF distance involves fitting the observed distribution of apparent [O iii] λ5007 magnitudes to a function that captures the PNLF's truncation, such as

$$\begin{eqnarray}&&N(M)\propto {e}^{0.307M}\{1-{e}^{3({M}^{* }-M)}\},\end{eqnarray} \tag{ 2 }$$

where M *= −4.54 is the [O iii] λ5007 absolute magnitude of the brightest possible planetary. This procedure proved extremely successful for galaxies within ∼10 Mpc, but beyond this distance, the values obtained from the PNLF were generally ∼0.2 mag smaller than those from Cepheids and the SBF method. This offset suggested the presence of one or more systematic errors (e.g., Ferrarese et al. 2000; Ciardullo 2022), such as the inclusion of contaminating objects (i.e., SNRs, H ii regions, and background galaxies) in the PN samples, or an incorrect expression for the shape of the PNLF cutoff.

The MUSE analyses by Spriggs et al. (2021) and Scheuermann et al. (2022) have pointed to another possible error in the PNLF: the projection of two separate PNe onto a single spatial (and spectral) resolution element. Although a chance alignment of two rare objects would seem improbable, Paper I demonstrated that photometric blends occur more often than previously realized. When this happens, a source may appear to have the spectrum of a normal PN, but with an [O iii] magnitude that is up to 0.75 mag brighter than M*. This can distort the observed PN luminosity function and lead to an incorrect (underestimated) value for both the PNLF distance and its uncertainty.

Chase et al. (2023) detailed the procedures needed to include the possibility of image blends in a PNLF analysis. In brief, if the [O iii] fluxes of two superposed PNe are uncorrelated, the probability density function (PDF) for their summed [O iii] flux is simply the convolution of the single object PDF with itself. At any position in a galaxy, the probability of observing a PN with a given magnitude m is therefore given by the weighted sum of two PDFs: one for single objects (Equation 2) and one for superpositions. Under the assumption that the distribution of PNe within a galaxy follows that of the underlying light, the weights of the two PDFs can be found by applying Poisson statistics to the number of PNe expected to be found on a single resolution element of data. This expectation value can then be estimated from the surface brightness of the galaxy, the total number of PNe observed, the velocity dispersion of the galaxy's stars, the image quality of the data, and the spectral resolution of the instrument.

Note that in our analysis, the expected PDF for every PN candidate depends not only on the form of the empirical function (Equation (2)) but also on the galactic surface brightness and velocity dispersion underlying the PN's location. Thus, one cannot simply fit the data with a single curve: a galaxy's best-fit distance must be derived by maximizing the combined likelihoods of the entire PN sample based on their individual PDFs. When PN blends are a possibility, histograms and cumulative distributions of PN magnitudes are useful visualization tools, but should not be used as the basis for a quantitative analysis.

As stated in the previous two paragraphs, a PNLF distance analysis, which includes the possibility of blends requires having some idea of the surface brightness distribution of the galaxy, the velocity dispersion of the stars underlying the position of each PNe, and the spectral and spatial resolution of the data (i.e., how close can two PNe be before we cannot tell there are two objects at one location). These values do not necessarily need to be precise, as there is some degeneracy between the parameters. However, PNLF analyses that do not include these factors can produce distances (and associated uncertainties) that are underestimated.

The most important parameters for modeling the effect of blends on the PNLF are the total amount of galaxy light sampled by a PN survey and the galactic surface brightness underlying the position of each PN. These data can either be estimated from previously published surface photometry, or from continuum measurements off the MUSE data cubes themselves. Of secondary importance is the stellar line-of-sight velocity dispersion at each PN's position in the galaxy. Such data are not available for all galaxies, and even then, the precision of the measurements is generally limited. Fortunately, PNLF distances are generally not very sensitive to this parameter, and any error in this quantity is easily subsumed into the overall error budget of the calculation. Finally, any analysis that includes the possibility of PN superpositions must include the spatial and spectral resolution of the data cube at the wavelength of the observed emission line. Simulations performed by Chase et al. (2023) show that to a good approximation, two adjacent PNe can only be spatially resolved in a MUSE data cube if their angular separation is greater than half the seeing FWHM or their velocity difference is greater than Δv ≈ 200 km s⁻¹.

NGC 253 is an SAB(s)c starburst galaxy and the most massive member of the Sculptor group. At the time of our analysis, two MUSE data cubes of the galaxy's central regions were available in the ESO archive (IDs ADP.2018-11-22T21:29:46.157, ADP.2019-08-24T09:53:08.548, PI: L. Zschaechner, Program ID 0102.B-0078); a much more comprehensive set of observations has since been released and will be the subject of a separate paper. The exposure times of the two data cubes considered here are similar (1820 s for P1 and 1833 s for P2), and our measurements of the image quality at 5007 Å are similar (106 and 095 for P1 and P2, respectively). The locations of these pointings are shown in Figure 1.

The only previous PNLF study of NGC 253's PNLF is by Rekola et al. (2005), who used the classical on/off-band imaging technique to find 14 PNe in the field outlined in gray in the right panel of Figure 1. These authors also modeled the effect of dust extinction in the galaxy; while one generally does not expect the PNLF to be greatly affected by galactic internal extinction—the scale height of PNe should be several times that of the dust (e.g., Feldmeier et al. 1997; González-Santamaría et al. 2021; Guo et al. 2021)—the high inclination of NGC 253 (i = 74°) and the galaxy's prominent dust lanes are clearly an issue. Rekola et al. (2005) concluded that the effect of extinction on their sample of NGC 253 PNe is quite small, and derived a PNLF distance to the system of ${(m-M)}_{0}={27.62}_{-0.26}^{+0.16}$ for a foreground reddening of E(B − V) = 0.019.

Our initial list of pointlike [O iii] sources in NGC 253's P1 and P2 fields contained 48 and 42 objects, respectively. However, after excluding objects based on their spectra (i.e., eliminating H ii regions and SNRs via the line strengths of Hα and [S ii]), the list of PN candidates dropped to 19 and 15 sources. Unfortunately, less than a dozen of these objects have magnitudes in the top ∼1 mag of the PN luminosity function.

The derivation of NGC 253's PNLF distance presents a number of challenges. The first comes from the diffuse ionized gas (DIG) distributed throughout the field. This gas, along with the galaxy's H ii regions and SNRs, is responsible for the large number of interlopers in our original list of PN candidates (red circles in the left panels of Figure 1) and for the possible existence of systematic errors in the background subtraction. As a result, the relatively small formal photometric errors associated with the brightest PNe (∼0.01 mag) do not necessarily account for the true uncertainties in our measurements. To compensate for this, we included an additional 5% error in all our photometry.

A second issue arises from the possibility of internal galactic extinction. As evidenced by their Balmer decrements, many of the PNe in NGC 253 are heavily obscured (e.g., Figure 2). Simplistic scale-height-based arguments (e.g., Feldmeier et al. 1997; González-Santamaría et al. 2021; Guo et al. 2021) suggest that internal extinction should not affect most PNLF measurements, but if NGC 253's dust is extincting the PNe, the effect should be removed before deriving a distance. Alternatively, since NGC 253 is a starburst object, it is also likely that many of its PNe have evolved from relatively high-mass progenitors. The circum-nebular extinction produced by these objects should not be touched since the component is an important contributor to the location of the PNLF cutoff (Ciardullo 2012; Davis et al. 2018). Unfortunately, since there is no easy way to disentangle the two sources of extinction, one must simply live with the possibility that internal galactic extinction, might be affecting the results. If so, then our formal PNLF distance will be an overestimate. (This argument applies to all the dusty galaxies analyzed in this paper.)

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Finally, we note that the P1 and P2 fields of NGC 253 do not possess any bright reference stars with which to determine the observations' aperture corrections. This is not an uncommon issue with MUSE archival data, since the field of view of the instrument is only $1^{\prime} \times 1^{\prime}$ (see Paper I). Our aperture corrections were therefore estimated using bright, compact star clusters as a reference. For distant galaxies, the use of such systems for point-source photometry should be valid, but at the distance of the Sculptor group, some of the clusters may be marginally resolved. If so, their use as point sources could introduce a zero-point error into our photometry and artificially brighten the PNLF. Additionally, since there is little overlap between the two MUSE fields, the errors associated with the aperture corrections may result in the photometry of the two fields having different zero-points. The systematic error introduced by this effect is discussed in Section 5.10.

Since our PN sample contains only ∼10 objects in the top magnitude of the luminosity function, a plot showing the differential distribution of these points is not very revealing. Consequently, Figure 3 plots the cumulative distribution of the PN magnitudes. (We will present cumulative distributions in later cases, as well, when the PN count is low.) The rapid departure from the distribution predicted by Equation (2) beyond m₅₀₀₇ ∼ 26.6 is not principally due to incompleteness. Rather, it comes from a dip in the luminosity function ∼1.5 mag below the bright-end cutoff. This feature, which was first reported by Jacoby & De Marco (2002) for the PNLF of the SMC, is characteristic of most star-forming systems, though the exact location and strength of the dip varies with stellar population. (For a qualitative explanation of the phenomenon, see Ciardullo 2010.) At brighter magnitudes, the PN magnitude distribution follows the predictions of the empirical law quite well (i.e., a Kolmogorov–Smirnov (K-S) test cannot rule out the hypothesis that the PNe are drawn from that distribution), but with so few objects in the top ∼1 mag of the PNLF, the exact position of the function's cutoff cannot be fixed with any precision.

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We used the analysis program of Chase et al. (2023) to derive NGC 253's PNLF distance. To estimate the likelihood of any PN superpositions, we used the H-band surface photometry of Forbes & Depoy (1992) to measure the amount of galaxy light at each position in the data cube and converted these IR magnitudes to the V band using the galaxy's integrated optical and IR colors (Aaronson 1977; Jarrett et al. 2003). We then assumed that the spectral lines of any possible PN blend would be unresolved, as the line-of-sight velocity dispersion in a disk galaxy such NGC 253 should be much less than MUSE's spectral resolution. We then computed the galaxy's most likely distance modulus, under the assumption of a foreground Milky Way reddening of E(B − V) = 0.016 (Schlafly & Finkbeiner 2011).

Figure 4 shows the results of our analysis. With so few objects in the top magnitude of the luminosity function, the random error associated with our solution is substantial. If we fit NGC 253's PNLF down to a limiting magnitude of m₅₀₀₇ = 26.6, we obtain a galaxy distance modulus of ${28.66}_{-0.28}^{+0.12}$, or ${5.4}_{-0.6}^{+0.3}$ Mpc. This number is about a magnitude more distant than the PNLF distance found by Rekola et al. (2005) from a different set of PNe identified at larger radii in the galaxy. (Unfortunately, there are no PNe common to both data sets.) It is also a magnitude further than the galaxy's TRGB distances of ∼27.5 (e.g., Mouhcine et al. 2005; Dalcanton et al. 2009; Radburn-Smith et al. 2011).

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The errors quoted above do not include the possible systematic offsets introduced by our very limited choice of aperture correction reference stars and our general lack of knowledge of the internal reddening. The former number is difficult to quantify with the present data set, while evidence for the latter is indirect. Although Rekola et al. (2005) concluded that internal extinction likely did not strongly affect their sample of NGC 253 PNe, our MUSE pointings lie much closer to the center of the galaxy, where the extinction is likely to be larger. Moreover, the best-fit value of α_2.5 displayed in Figure 4 is slightly lower than expected for PN measurements in a spiral galaxy (Ciardullo 2010), suggesting that dust may be affecting PN detections.

The two MUSE data cubes analyzed in this work cover only a small fraction of the galaxy's ∼3 × 10¹⁰ L_⊙ of the B-band light. Thus, our result is just a feasibility test. However, since 2022 a more complete survey of NGC 253 has been published in the ESO archive. These MUSE pointings include proper aperture correction stars and should contain enough PNe to produce a PNLF statistical error comparable to that of the galaxy's TRGB distance measurement. We will address this opportunity in a future paper.

NGC 1052 is a bright elliptical (E4) galaxy with a radio jet, a LINER-type nucleus, and a large-scale outflow that has been studied with the Wide-Field IFU spectrograph on the Australian National University 2.3 m telescope (Dopita et al. 2015). Recently, the system has garnered considerable attention, as two of NGC 1052's satellites, DF2 and DF4 appear to be ultradiffuse dwarf galaxies with little or no dark matter (van Dokkum et al. 2018). Since these dark matter estimates depend on the system's distance, an independent measure of this quantity is of considerable interest.

The ESO archive provides a data cube derived from the merger of two MUSE pointings with an effective exposure time of 1685 s and a measured seeing at 5007 Å of 080 (ESO archive ID ADP.2019-10-05T19:19:48.724, PI: L. Hernandez-Garcia, Program ID: 0103.B-0837). As shown in Figure 5, the data span the galaxy's minor axis, and overlap in the area of its bright nucleus The diff image, shown in Figure 5 highlights the outflow, which is quite bright, even in the high-excitation line of [O iii] λ5007.

Our initial examination of NGC 1052's MUSE data cube found 100 PN candidates, with the brightest 20 having photometric errors less than 0.05 mag. However, we rejected 13 of these candidates, as their spectra are inconsistent with that of an [O iii]-bright PN (e.g., Herrmann et al. 2008; Kreckel et al. 2017). These are identified in Figure 5 with red circles. In addition, as the distribution of objects in Figure 5 demonstrates, PN detections in the galaxy's inner 11'' are problematic, due to the extremely bright emission from the region's diffuse gas. The exclusion of this region removed one object from our sample, leaving 86 PNe suitable for analysis, and ∼50 in the critical top ∼1 mag of the PNLF.

One major challenge associated with measuring NGC 1052's PNLF is contamination by the galaxy's diffuse emission-line gas. As illustrated by the difference image of Figure 5, the DIG in NGC 1052 creates a bright, complex background that compromises the photometry of faint PNe, even in the high-excitation [O iii] line. Fortunately, in most cases, the PN's radial velocity is different enough from that of the DIG that the two components can be deblended quite easily using the MUSE spectra (see Paper I). The exception is in the nuclear region of the galaxy, where the luminosity of the outflow overwhelms that of the point sources.

The importance of NGC 1052's DIG is illustrated in Figure 6. The bottom panel shows the galaxy's PNLF as measured using simple point-source photometry, where the background is estimated using an annulus surrounding each source. The top panel shows the PNLF where the contribution of the diffuse emission-line gas has been removed by carefully examining each object's spectrum and deblending the PN's [O iii] emission from that of the DIG using the methodology of Paper I. Clearly, the latter set of photometry is a better fit to Equation (2) and produces a distance that is ∼0.25 mag more distant. This highlights another advantage of MUSE PNLF measurements: narrowband photometry would have been unable to disentangle the two sources of [O iii] emission.

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To fit NGC 1052's PNLF, we determined the amount of galactic V-band light at every position in the MUSE data cube using the R-band surface photometry of Jedrzejewski et al. (1987) and an assumed color of (V − R) = 0.93 (Persson et al. 1979). We find that, after excluding the central 11'' of the galaxy, roughly V ∼ 11.1 of flux is contained within the two MUSE pointings. We then estimated the stellar velocity dispersion underlying each PN's position using the long-slit spectroscopy of Binney et al. (1990), and computed the most likely distance to the galaxy assuming a foreground extinction of E(B − V) = 0.023 (Schlafly & Finkbeiner 2011). Our solution using these numbers is illustrated in Figure 7.

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As shown in the figure, Equation (2) fits the observed PNLF extremely well, and there is no evidence of any overluminous objects. Our PNLF distance to NGC 1052 is ${(m-M)}_{0}={31.26}_{-0.07}^{+0.04}$ (${17.9}_{-0.6}^{+0.3}$ Mpc), where the uncertainties do not include the systematic error associated with MUSE's flux calibration and our photometric aperture correction. The former uncertainty is generally of the order of ∼3% (Weilbacher et al. 2020); to estimate the latter, we examined the photometry of the four, pointlike sources contained in the overlap region of the two NGC 1052 MUSE fields. These sources are relatively faint, with between 17,175 and 37,780 counts (in the units as explained in Appendix B), and the scatter of their inferred aperture corrections is σ = 0.075 mag. If we use the standard deviation of the mean to define the expected error on the data cube's aperture correction, then this systematic component is ∼0.04 mag.

(We do note that the pointlike sources used to define the data cube's PSF are likely not Milky Way stars, but globular clusters belonging to NGC 1052. However, this should not affect our measurement of the aperture correction. If we scale M31's globular cluster system (Barmby & Huchra 2001) to the distance of NGC 1052, then a median globular cluster in the galaxy should have a half-light radius of only 002, and the angular size of the M31's largest cluster would be R_e ∼ 031. This is much smaller than the 080 seeing of the data cube.)

If we fold in the systematic errors associated with the MUSE flux zero-point, the data cube's aperture correction, and the likely error on the reddening, then our estimate for the galaxy's distance modulus becomes ${(m-M)}_{0}={31.26}_{-0.08}^{+0.07}$ (for an M* value of −4.54). This distance is essentially identical to that found from the Hubble Space Telescope (HST)/NICMOS measurement of the galaxy's near-IR SBFs ((m − M)₀=31.28 ± 0.27; Jensen et al. 2003). It is also consistent with the distances of the galaxy's two ultradiffuse satellite dwarfs, as inferred by first taking the ratio of their I-band SBFs to those of the dwarf galaxies of the M96 group, and then anchoring the zero-point to the megamaser distance of NGC 4258 (van Dokkum et al. 2018). Interestingly, direct TRGB measurements of these dwarfs give significantly larger values for the group's distance, with (m − M)₀ = 31.72 ± 0.12 (Shen et al. 2021) and 31.50 ± 0.18 (Danieli et al. 2020). These distance moduli are excluded by our measurements. Finally, we note that Fensch et al. (2019) report the identification of three PNe in the dwarf NGC 1052-DF2. However, since none of these objects are likely to have magnitudes near the bright-end cutoff of the PNLF, their detection does not provide a useful constraint on the galaxy's distance.

NGC 1326, otherwise known as Fornax Cluster Catalog (FCC) galaxy 029 (Ferguson 1989), is a bright lenticular system (Hubble type (R1)SAB(r)0/a) with an outer ring of star formation (roughly 2' from the nucleus) and an inner ring-like structure extending ∼9'' in diameter. Although the galaxy has Tully–Fisher measurements in the range of ∼13 to ∼19 Mpc (e.g., Bottinelli et al. 1984; Willick et al. 1997; Springob et al. 2009), the system's early Hubble type and moderately face-on (i ≈ 45°) inclination calls those values into question. Nevertheless, the system's location in the core of the Fornax cluster, 31 from the central cD galaxy NGC 1399, suggests that the Tully–Fisher estimates are reasonable. There are no Cepheid, TRGB, or SBF distances to the system.

We retrieved from the ESO archive a MUSE-DEEP data cube formed from the combination of two exposures (ESO Archive ID: ADP.2018-04-05T08:26:13.290, PI: M. Carollo, Program ID: 0100.B-0116). The effective exposure time for these data is 3354 s, and the seeing at 5007 Å is 100. The footprint of the IFS is shown in Figure 8. The region encompasses the system's bright nuclear region and much of the galactic bar.

PN detections within the central 12'' of NGC 1326's nucleus are difficult, due to the region's high surface brightness and the copious amounts of diffuse [O iii] emission. Thus, we confined our analysis to larger galactocentric radii, where we found 86 pointlike emission-line sources. Many of these objects, especially those close to the nucleus and along the galactic bar (the red circles in Figure 8), have the spectral signature of H ii regions or SNRs and are excluded from our list of PN candidates. The final sample therefore consists of 55 objects; this PNLF is shown in Figure 9.

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NGC 1326's PNLF is not very well defined. In part, this is because there is one PN candidate that is 0.31 mag more luminous than any other object. (Figure 9 shows the PNLF with the starting edge of the first bin chosen to minimize the discordant appearance of PN1. ⁶ ) However, a more important factor is that, of the 40 PNe brighter than the nominal completeness limit of m₅₀₀₇ = 27.8, only ∼20 are in the magnitude range that defines the PNLF cutoff (m ≲ M* + 1).

There are several possible explanations for the overluminous nature of PN1. The first, and simplest, is that the apparent luminosity of the object (PN1) is simply due to poor photometry. As can be seen in the diff image in Figure 8, PN1 is located in a region of bright diffuse emission-line gas, adjacent to an interloping object (red circle). It is quite possible that the source's apparent high luminosity is due to imperfect background subtraction, in which case the object should be excluded from the analysis. Similarly, if the overluminous object is something other than a normal PN (such as a high-excitation, metal-poor H ii region), it should also be eliminated from the data set. There is no evidence for this in the object's spectrum, but the interpretation cannot be completely excluded.

A third explanation is that there is nothing abnormal about the PN at all, and the 0.31 mag difference between it and the next brightest PNe is simply due to small number statistics. This seems unlikely, as that would imply that NGC 1326 is at least ∼4 Mpc closer than the main body of Fornax. But if the object is a normal PN, then it must be included in the fitting process. Similarly, it is possible that this bright [O iii] source is actually formed from the combined flux of two normal PNe that are superposed upon each other. To wit, PN1 is located in a region of NGC 1326 with a V-band surface brightness of μ_V ∼ 19.7 mag arcsec⁻² (Buta et al. 1998). According to Equation (3) of Chase et al. (2023), this means at the nominal Fornax distance of 19 Mpc, PN1 has a ∼20% chance of being a superposition of two objects in the top 2.5 mag of the PN luminosity function, and a ∼5% chance that both PNe are in the top 1 mag of the PNLF. If this hypothesis is correct, then, once again, the source must be included in the PNLF analysis.

Finally, it is possible that PN1 is a normal PN, but that our knowledge of the true shape of the PNLF is incomplete. Equation (2) was originally defined to fit the magnitude distribution of a statistically complete sample of ∼120 PNe in the bulge and inner disk of M31 (Ciardullo et al. 1989). In such a finite data set (with only ∼32 objects in the top ∼1 mag of the luminosity function), extremely rare, short-lived objects may not be present. Indeed, although [O iii] PNLFs now exist for dozens of galaxies, the PN samples are rarely large enough to exclude the possible existence of a very low-amplitude, bright-end tail to the luminosity function. If such a tail does exist, any attempt to fit PN1 with the expression given by Equation (2) will result in a systematic underestimate of the galaxy's distance. We will discuss the true shape of the empirical PNLF in Section 6.3.

We fit the PNLF of NGC 1326, with and without PN1, using the photometric measurements of Buta et al. (1998) to determine the galaxy surface brightness underlying each PN candidate. These data imply that the total amount of V-band light contained within the MUSE data cube is V ∼ 11.5. However, we made no attempt to model the behavior of the stellar velocity dispersion in the MUSE field: since the spectroscopy of Dalle Ore et al. (1991) and Gadotti & de Souza (2005) both indicate that the line-of-sight velocity dispersion is much less than the velocity resolution of the MUSE spectrograph, we set σ = 100 km s⁻¹ throughout the survey region. Finally, we assume E(B − V) = 0.016 as the foreground extinction to the galaxy (Schlafly & Finkbeiner 2011).

Figure 10 shows the results of our fits. As expected, the galaxy's distance is not well defined: 23 objects spread out over the top ∼1 mag of the PNLF is insufficient for defining the precise location of the PNLF cutoff. As a result, whether or not one includes PN1 in the sample does not substantially change the quality of the fit. If PN1 is treated as a normal PN, then the galaxy is forced to a distance that is ∼4 Mpc smaller than the canonical distance to Fornax, i.e., to ${14.6}_{-0.7}^{+0.4}$ Mpc. This result is unaffected by the inclusion of superpositions in the analysis: although it is possible that the object is a superposition of two sources, the likelihood that two PNe in sum would produce an [O iii] source ∼0.3 mag brighter than the next most luminous object is quite low and barely registers in the contour plot.

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The exclusion of PN1 from consideration produces a fit that is only slightly better than that which includes the PN. (Without PN1, the K-S statistic for a comparison of the data with the model is D_n = 0.162, for a p-value of p = 0.77; with PN1, D_n =0.174, and p = 0.83.) Yet without PN1, the galaxy's most likely distance modulus is Δμ = 0.19 mag greater, ${(m-M)}_{0}={31.00}_{-0.11}^{+0.06}$, or $d={15.9}_{-0.8}^{+0.5}$ Mpc.

The uncertainties quoted above do not include the systematic errors associated with the reddening, MUSE flux calibration, data cube aperture correction, or limitations in the reference PNLF. In particular, although the aperture correction error obtained from the field's brightest pointlike source has a calculated uncertainty of only 0.028 mag, the standard deviation about the mean aperture correction for the three brightest PSF objects is 0.085 mag, implying an 0.05 mag error of the mean. This uncertainty, along with those for the MUSE flux calibration and foreground reddening, results in a distance modulus of ${31.00}_{-0.13}^{+0.09}$, or $d={15.9}_{-0.9}^{+0.6}$ Mpc. This number is still significantly smaller than the distance to Fornax, and suggests that the system may be foreground to the cluster's core.

NGC 1351 (FCC 083) is another moderately bright Fornax lenticular (classification SAOp) that has two pointings in the ESO archive (ESO Archive IDs: ADP.2017-12-12T15:38:27.863 and ADP.2017-07-19T15:12:54.145, PI: M. Sarzi, Program ID: 296.B-5054). The first pointing, P1, is centered on the nucleus and has a nominal seeing at 5007 Å of 088 and an exposure time of 3379 s; the second, P2, overlaps P1, as it is centered ∼35'' northwest of the nucleus along the galaxy's major axis. The latter data cube has a somewhat longer exposure (4680 s) and slightly worse (111) image quality.

Our examination of the central pointing of NGC 1351 found 92 PN candidates, of which six were later rejected as interlopers based on their spectra. P2 had far fewer objects, as it sampled less light with poorer image quality: this pointing only added 16 PNe to the sample. In total, we detected 102 PNe in NGC 1351, with more than 30 objects in the top ∼1 mag of the luminosity function.

The observed PNLF is shown in Figure 12. The displayed data points exclude two objects (PN8 and 32) located within 10'' of the nucleus, as, in this region, the galaxy's bright background severely limits PN detections. But the remaining objects outline a fairly normal PNLF. In fact, the overall distribution is quite similar to that of NGC 1326, though with a greater population of PNe in the top ∼1 mag of the luminosity function. Like NGC 1326, NGC 1351's PNLF contains one object whose spectrum is fully consistent with that of normal PNe, but whose luminosity is 0.2 mag brighter than expected from Equation (2). Again, there are three explanations for the object.

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The first is that the apparent luminosity of PN1 may be due to poor photometry. PN1's [O iii] flux is partially blended with that of PN16, which is projected less than 1'' away (see Figure 13). Since the radial velocities of the two objects are similar, imperfect deblending of their PSFs could result in PN1 appearing overluminous. The second is that the object is some form of PN (or PN mimic) that does not obey the luminosity function defined by Equation (2). This possibility will be addressed further in Section 6.2. Finally, the object may be an otherwise normal PN that is superposed on another object. The background upon which PN1 is projected is not particularly bright (μ_V ∼ 23.4 mag arcsec⁻²), but the possibility cannot be entirely discounted. Thus, we are left with the same ambiguity seen in NGC 1326, with one object having an outsized influence on the galaxy's derived distance. The difference between the two galaxies is that the larger number of PNe present in NGC 1351 makes the anomalous luminosity of PN1 more obvious. Consequently, if one assumes that PN1 is not an unlucky superposition of two sources, a K-S test can exclude its membership in the distribution defined by Equation (2) with greater than 93% confidence.

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The situation is summarized in Figure 14, which shows the results of fitting Equation (2) to the data with and without the brightest PNe. For the analysis, we calculated the likelihood of unresolved PN superpositions using the B-band galaxy surface photometry of de Carvalho et al. (1991), a mean color of (B − V) = 0.90 (Faber et al. 1989) and the velocity dispersion data of D'Onofrio et al. (1995). ⁷ From the figure, it is clear that the calculated distance to the galaxy depends on whether PN1's photometry is accurate and whether it is, indeed, a normal PN, rather than some exotic object.

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If PN1 is treated as a normal PN, then the galaxy's inferred distance modulus is ${(m-M)}_{0}={31.23}_{-0.07}^{+0.04}$ (${17.7}_{-0.6}^{+0.3}$ Mpc) for E(B − V) = 0.011. This is a formal solution only: as stated above, a K-S test excludes the null hypothesis that the observed set of PNe is drawn from the distribution defined by Equation (2). On the other hand, if the object is excluded from the sample, the galaxy's distance modulus increases to ${(m-M)}_{0}={31.39}_{-0.08}^{+0.04}$ (${19.0}_{-0.7}^{+0.4}$ Mpc), and the quality of the fit is much improved. Consequently, we adopt this larger distance in our analysis.

For NGC 1351, the aperture correction measurements are quite good, and the total systematic error, due to uncertainties in the MUSE flux calibration, the data cubes' aperture corrections, and foreground extinction, is ∼0.06 mag. This increases the error bars slightly so that the distance modulus to the galaxy becomes ${(m-M)}_{0}={31.39}_{-0.10}^{+0.05}$, or ${19.0}_{-0.9}^{+0.7}$ Mpc. This value is nearly identical to the z₈₅₀-band SBF distance of 31.42 ± 0.07 (19.2 ± 0.6 Mpc) (Blakeslee et al. 2009).

One other feature of Figure 12 is worth noting. The number of PNe observed, normalized to the total amount of galaxy light contained in the MUSE data cubes (found using the surface photometry of de Carvalho et al. 1991), is higher than that expected for early-type galaxies. This anomaly does not substantially affect the implied distance modulus to the galaxy, since the methodology of Chase et al. (2023) only requires knowledge of the relative amount of light at each position in the MUSE data cube. But it suggests that there is either a zero-point issue associated with the galactic surface photometry or that the stellar population of NGC 1351 is a bit younger than that of most lenticular systems.

NGC 1366 is an edge-on S0 galaxy in the Fornax cluster with an absolute magnitude of M_I = −19.78 and evidence of a counterrotating core (Morelli et al. 2017). The ESO archive contains a single deep data cube of the galaxy (ESO Archive ID: ADP.2019-10-10T08:04:58.194, PI: L. Morelli, Program ID: 0103.B-0331) with an effective exposure time of 2446 s and an image quality of 112 at 5007 Å.

Our DELF procedure reveals a large amount of high-excitation ([O iii] λ5007) line emission in the galaxy's inner ∼15''. As can be seen in the diff image in Figure 15, this light has the appearance of a spiral feature that is oriented perpendicular to the plane of the disk. Given the velocity structure of this gas (Morelli et al. 2017), it is hard to imagine that this morphology is the result of an outflow similar to that seen in NGC 1052 (Section 5.2). However, the feature does resemble those seen in Illustris TNG100 cosmological simulations of galaxies with counterrotating cores and infalling gas (Khoperskov et al. 2021). Given that NGC 1366 is known to have counterrotation, the models seem to be a plausible explanation for the appearance of the gas.

Figure 15 also identifies a number of pointlike [O iii] sources superposed on the diffuse emission. Owing to the modest luminosity of the host galaxy and the data cube's poor image quality, we initially detected a set of only 31 PN candidates. Upon further inspection of the spectra, this number was pared down by 12, leaving just 19 objects in our PN sample, and only 13 in the brightest ∼1 mag of the luminosity function.

Figure 16 shows the cumulative distribution of PN magnitudes in NGC 1366. Clearly, the distribution is in good agreement with the empirical law of Equation (2). But it is also clear that with only 18 objects in the statistically complete sample, and just 13 in the critical top magnitude of the luminosity function, the exact location of the PNLF cutoff is poorly defined.

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We analyzed the galaxy's PNLF using the R-band surface photometry of Morelli et al. (2008) with an assumed (V − R) color of 0.55 (Prugniel & Heraudeau 1998). These data imply that, outside the galaxy's central 10'', where no PNe are detected, the MUSE data cube surveys V ∼ 12.5 of galaxy light. We also assumed a line-of-sight stellar velocity dispersion of 100 km s⁻¹ throughout the region of PN detections; this is consistent with the spectroscopy of Morelli et al. (2017). The results of our analysis are displayed in Figure 17.

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With a reddening of E(B − V) = 0.014, the formal PNLF fit for NGC 1366 gives ${(m-M)}_{0}={31.39}_{-0.22}^{+0.10};$ if we assume that flux calibration, aperture correction, and reddening contribute an additional ∼0.06 mag to the error budget, the final distance modulus becomes ${(m-M)}_{0}={31.39}_{-0.23}^{+0.11}$, or ${19.0}_{-1.8}^{+1.0}$ Mpc. This value is consistent with Fornax cluster membership and is in accord with the I-band SBF distance modulus of (m − M)₀=31.62 ± 0.29 found by Tonry et al. (2001). We note that the value for the PNe per unit luminosity value is slightly low for a galaxy of this type. However, given the small number of PNe detected and the possibility that we are losing objects due to the bright, high-excitation line emission in the galaxy's core, the estimate of α is reasonable.

NGC 1385 is an Eridanus cluster barred spiral galaxy (Hubble type SB(s)cd) with publicly available data products in the ESO Archive from the PHANGS-MUSE survey (Emsellem et al. 2022, PI: E. Schinnerer, Program ID: 1100.B-0651). The galaxy is particularly interesting for a distance scale study since it has no TRGB, SBF, or Cepheid measurements, and its Tully–Fisher distances extend over a very wide range, from ∼8 Mpc (e.g., Bottinelli et al. 1986; Sorce et al. 2014) to ∼50 Mpc (Bottinelli et al. 1984). Figure 18 shows the five pointings in the MUSE archive, which cover most of the galaxy's light. These pointings, along with their exposure times and seeing at 5007 Å, are listed in Table 4.

Table 4. Data Cubes for NGC 1385

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
P1	ADP.2020-01-11T01:24:00.728	2420	064
P2	ADP.2021-01-29T13:19:05.970	2420	077
P3	ADP.2020-01-28T16:14:42.084	2420	082
P4	ADP.2019-02-13T01:31:18.894	2420	071
P5	ADP.2020-01-27T19:05:08.471	2420	064

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We note that P1, P4, and P5 were observed twice; however, the combined data cubes are not available in the ESO archive. In addition, the archive also contains two mosaicked data cubes (ESO Archive IDs: ADP.2021-07-16T10:20:56.387, ADP.2021-07-16T10:20:56.381). The cube that we label as A1 was created by merging all of the individual data sets into one cube, including the fields with two visits. The second data cube, A2, also combines all the data, but only after convolving each individual cube to a common PSF with a 077 FWHM. This process is described in Emsellem et al. (2022).

To identify the galaxy's PN candidates, we first examined the individual data cubes (P1–P5), and one by one, found and measured 24, 23, 16, 4, and 19 PN candidates, respectively. We then repeated the photometry in mosaic A2, which has greater depth in the regions P1, P4, and P5, due to the fields' second observation. The improved signal-to-noise ratio (S/N) allowed us to exclude eight of our initial PN candidates as interlopers. Our final sample of candidates therefore contains 78 objects, with 54 being in the top ∼1 mag of the luminosity function.

The positions of these PNe are shown in Figure 18. From the figure, it is clear that PN detections within ∼30'' of the nucleus and along the galaxy's bright northern arm are severely compromised by the galaxy's high surface brightness and bright line emission. We exclude these regions from our analysis and thus eliminate one additional object (PN9) from our sample.

Figure 19 shows the luminosity function of the remaining PN candidates. Despite all the active star formation and dust present in the galaxy, the bright end of NGC 1385's PNLF is extremely well fit by the empirical law. This is in agreement with expectations: because the scale height of PNe should be larger than that of the galactic extinction, the bright end of the PNLF in large, moderately face-on galaxies should be dominated by objects on the nearside of the galaxy, above the dust layer (Feldmeier et al. 1997; Rekola et al. 2005). There is also no evidence of any overluminous PNe in the system.

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Since there are no surface photometry measurements over the face of NGC 1385, we estimated the amount of light underlying each PN and in the entire MUSE data cube using continuum measurements made directly from the MUSE spectra. Also, since NGC 1385 is a disk galaxy seen relatively face-on (inclination angle of ∼44°), the line-of-sight velocity dispersion of its stars should be much less than the galaxy's ∼136 km s⁻¹ rotation speed (Lang et al. 2020). Thus, we can expect superposed PNe to be unresolved, and adopt ∼50 km s⁻¹ as a typical value for the system's line-of-sight velocity dispersion.

Figure 20 shows the result of fitting the observed PNLF to Equation (2). For a Milky Way foreground extinction of E(B − V) = 0.017 (Schlafly & Finkbeiner 2011), the fitted distance to NGC 1385 is ${(m-M)}_{0}={31.99}_{-0.08}^{+0.06}$.

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Unfortunately, obtaining the true error bars for the NGC 1385's distance is difficult. Pointings P2 and P5 have reasonably bright point sources in the field, and estimates for their aperture corrections are consistent to within 0.03 and 0.04 mag, respectively. However, in fields P1, P2, and P5, the PSF star photometry displayed large uncertainties with formal errors on the aperture corrections as large as 0.25 mag. It is highly unlikely that the true errors are as large as this since the effect would blur out the PNLF cutoff and create a luminosity function that is a poor match for Equation (2) (see Section 5.10). Conservatively, we adopt ∼0.1 as the total systematic error associated with our distance measurement. NGC 1385's distance is then ${(m-M)}_{0}={31.99}_{-0.12}^{+0.11}$, or ${25.0}_{-1.5}^{+1.4}$ Mpc. This places the galaxy well beyond the core of the Fornax cluster and in the mid-range of the Tully–Fisher estimates.

We note that NGC 1385 was included in the recent PNLF study by Scheuermann et al. (2022) that is based on the same data set that we have used. Their analysis resulted in a significantly shorter distance, with ${(m-M)}_{0}={29.96}_{-0.32}^{+0.14}$, or only ${9.8}_{-1.5}^{+0.6}$ Mpc. Interestingly, the authors caution that their derived distance modulus was subject to a large uncertainty and should be considered an upper limit to the galaxy's true distance. Specifically, they reported only 11 PNe brighter than their completeness limit of ${m}_{\mathrm{lim}}=28$, with the most luminous PN being brighter than m₅₀₀₇ = 26. This is not supported by our measurements, as our brightest PN has m₅₀₀₇ = 27.55. Also, the number of PNe contributing to our fit is almost 5 times larger than theirs.

Since the agreement of our PNLF distances for other galaxies in the Scheuermann et al. (2022) sample is quite good, we have looked into the possible causes of the discrepancy. A detailed account of the comparison is given in Appendix A. In summary, we note the following:

• (1)  
  Our sample of 77 [O iii] sources with spectra consistent with that of a PN extends ∼0.4 mag deeper than the Scheuermann et al. (2022) data set. This is readily understood from the superior capabilities of the DELF extraction, which typically yields a factor of ∼2 improvement in S/N over other techniques (see Paper I). From their list of 11 candidates, only two are confirmed by our analysis.
• (2)  
  The majority of the Scheuermann et al. (2022) PN candidates are located within giant H ii region complexes. This is problematic due to issues associated with background subtraction. With the exception of the two PNe mentioned above, we classify all these objects as H ii regions or SNRs. This explains why two-thirds of their objects have measured magnitudes brighter than our PNLF cutoff. Due to the difficulty of distinguishing H ii regions from PNe, our survey largely avoided those regions of the galaxy with the highest star formation rates.
• (3)  
  The short distance of 9.8 Mpc is not compatible with the SBF distances of other Fornax/Eridanus group galaxies (see Appendix A, Table 12). In fact, the system's radial velocity and our PNLF distance place NGC 1385 near the center of the Eridanus cloud (Willmer et al. 1989).
• (4)  
  The brightest object in the Scheuermann et al. (2022) sample is 1.83 mag more luminous than the brightest PN in our sample. Following the analysis of Soemitro et al. (2023), who examined the PNLF for NGC 300, another disk galaxy with active star formation, if the galaxy is at the distance of Eridanus, then their PN1 would need to be excited by a central star with a luminosity of at least $\mathrm{log}L/{L}_{\odot }=4.48$. This is much brighter than the post-AGB evolutionary track of any PN central star (Gesicki et al. 2018).

NGC 1399, the central cD/giant elliptical galaxy of the Fornax cluster, has been surveyed for PNe as far back as the early 1990s (McMillan et al. 1993; Arnaboldi et al. 1994). Two different MUSE programs have observed the galaxy (ESO Archive ID: ADP.2017-03-27T13:16:27.827, PIs: J. Walcher and S. Zieleniewski, Program IDs: 094.B-0903 and 094.B-0298) and these data have been combined into one large ($3\buildrel{\,\prime}\over{.} 92$ FoV) data cube with a quoted effective integration time of 954 s and an image quality of 081. In fact, according to the ESO archive provenance chain, this is a full-sized mosaic with 295 s exposure times everywhere, plus five 900 s exposures on the central regions that provide field overlap. This coaddition was not seamless: as the processed off and diff images of Figure 21 show the data suffer from imperfect flat-field corrections and different levels of noise. Nevertheless, our DELF technique allowed us to identify 232 PN candidates over the body of the galaxy.

Surface photometry of NGC 1399 exists from a number of studies (e.g., Franx et al. 1989; Caon et al. 1994; Iodice et al. 2016), and we used these data to determine the amount of V-band galactic light at every position in the MUSE data cube. For estimating the line-of-sight velocity dispersion at the position of each PN, we used the spectroscopy of Saglia et al. (2000), who obtained major and minor axis kinematic data over all but the outermost regions of the MUSE survey.

A comparison of the distribution of MUSE PNe and that of the galaxy's light reveals that we are likely missing PNe within 15'' of the galaxy's nucleus. This is not unexpected as the exposure times for this galaxy are relatively short. Consequently, the region's steep surface brightness gradient causes the limiting magnitude for PN detections to change rapidly with position. While we could perform artificial star experiments to model and incorporate the effect into our analysis, it is simpler to just exclude the entire region from our study. This reduces the PNe sample by less than 10% and still leaves over 100 PNe, covering V ∼ 10.3 mag of galaxy light, in the brightest ∼1 mag of the luminosity function.

Figure 22 shows the observed PNLF of the galaxy. Since the bright end of the function is so well populated, the shape of the PNLF is exquisitely defined. Moreover, despite the large number of PNe, there is no evidence of any overluminous objects. The lack of such objects can be partially explained by the galaxy's mass. NGC 1399 is a central cD galaxy whose line-of-sight velocity dispersion ranges from ∼370 km s⁻¹ near the nucleus to ∼250 km s⁻¹ in the outer regions surveyed by MUSE. Consequently, even when two PNe fall onto the same spatial element, there is a good chance that their fluxes can be disentangled via the objects' discordant radial velocities. Overluminous objects caused by PN superpositions should therefore be less common in such an environment, and agreement between the observed PN distribution and Equation (2) certainly supports this.

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Figure 23 shows the results of our maximum-likelihood analysis. If we assume a foreground reddening of E(B − V) = 0.011 (Schlafly & Finkbeiner 2011), then a fit of Equation (2) to NGC 1399's observed PNLF results in a distance modulus of ${(m-M)}_{0}={31.23}_{-0.05}^{+0.04}$. When folded in with the ∼0.05 mag zero-point uncertainty from the MUSE flux calibration, our determination of the data cube's aperture correction, and the uncertainty in the reddening, the inferred distance becomes ${(m-M)}_{0}={31.23}_{-0.07}^{+0.06}$ or ${17.6}_{-0.6}^{+0.5}$ Mpc.

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Both the distance and PN/luminosity values are almost identical to those measured by McMillan et al. (1993) using interference filter observations with the Blanco 4 m telescope through significantly worse (13) seeing. However, our value is almost 20% shorter than the latest HST-based SBF distance of 21.1 ± 0.7 Mpc from Blakeslee et al. (2010). It is also slightly closer than recent Cepheid and TRGB distances to the Fornax cluster, though not to NGC 1399 itself (e.g., Riess et al. 2016; Jang et al. 2018; Hoyt et al. 2021). However, the uncertainties in these measurements do largely overlap.

NGC 1404 is another Fornax galaxy, located near the very center of the cluster. For the long-term objective of our study, this E1 galaxy is particularly interesting, as it has hosted two Type Ia SNe, SN 2007on and SN 2011iv, whose distance estimates differ by as much as 14% (Gall et al. 2018). The galaxy is also compelling for technical reasons since a bright foreground star is projected onto the body of the galaxy. This star provides an excellent measure of the field's PSF.

There are four useful data cubes of NGC 1404 in the MUSE archive. Field P1, which includes the galaxy's nucleus, is a 3287 s exposure with 088 seeing at 5007 Å (ESO Archive ID: ADP.2017-12-13T01:47:07.213, PI: M. Sarzi, Program ID: 296.B-5054). The three other exposures are of a halo field (P2), which is located north–northwest of the nucleus along the system's major axis. The exposure times for these three cubes are 1680 s, and their net seeing is 101 at 5007 Å (IDs: ADP.2017-12-05T15:14:58.786, ADP.2017-12-01T13:57:32.491, ADP.2017-12-01T13:57:32.480; same PI and ID as above). As the archive did not provide a combined data cube for P2, we chose to reduce the entire data set anew. The locations of both fields are shown in Figure 24.

Our DELF analysis of NGC 1404's data cubes initially found 179 PN candidates, with the vast majority located in the central field. Further inspection of the candidates' spectra eliminated 53 objects from consideration, either on the basis of classification as interlopers, or as spurious detections that were too faint to meet the detection criteria. This left us with a sample of 126 PNe candidates: 107 in P1, 18 in P2, and one object common to both data cubes. (The two measurements of this object, 27.92 ± 0.05 on P1 and 27.88 ± 0.07 on P2 are in excellent agreement.) Most importantly, ∼64 of the PNe populate the top ∼1 mag of the luminosity function.

Like NGC 1399, NGC 1404 has been well studied photometrically (e.g., Franx et al. 1989; Sparks et al. 1991; Muñoz-Mateos et al. 2009) and kinematically (D'Onofrio et al. 1995; Iodice et al. 2019), allowing us to estimate the amount of light and line-of-sight velocity dispersion at every location in the data cube. Also like NGC 1399, a comparison of the distribution of light and PNe in the galaxy reveals that our survey is incomplete within ∼10'' of the nucleus. The exclusion of this region removes seven PNe from the sample and leaves V ∼ 10.8 mag of galaxy light in the two data cubes.

Figure 25 shows NGC 1404's PNLF, along with the best-fitting PN luminosity function. The figure illustrates that our PN detections in NGC 1404 do not go as deep as in some other galaxies of Fornax, and faint-end incompleteness sets in only ∼0.8 mag below M*. Nevertheless, the PNLF cutoff is very well defined, and for a foreground reddening of E(B − V) = 0.010, the analysis displayed in Figure 26 yields a distance modulus of ${(m-M)}_{0}={31.37}_{-0.07}^{+0.04}$. When combined with the ∼3% error on the MUSE flux calibration (Weilbacher et al. 2020), and the very small uncertainties associated with the aperture correction (see Appendix B) and reddening, the final distance modulus becomes ${(m-M)}_{0}={31.37}_{-0.08}^{+0.05}$, or ${18.8}_{-0.6}^{+0.4}$ Mpc.

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Unsurprisingly, as Figure 27 illustrates, our distance is in excellent agreement with the (m − M)₀ = 31.42 ± 0.1 value derived by Spriggs et al. (2020), as the measurements use the same MUSE data cubes and are not systematically different (although the error bars on the DELF photometry are smaller; see Paper I). Our value is also essentially identical to the galaxy's two TRGB distance moduli, (m − M)₀ = 31.36 ± 0.04 (stat)±0.05 (sys) measured by Hoyt et al. (2021), and 31.29 ± 0.07 derived by Anand et al. (2022). It is, however, significantly more distant than the poorer-seeing PNLF estimate of ${(m-M)}_{0}={31.15}_{-0.10}^{+0.07}$ found by McMillan et al. (1993).

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NGC 1419 is a small elliptical galaxy in the Fornax cluster, with SBF distances generally ranging from 19–22 Mpc (Blakeslee et al. 2001; Tonry et al. 2001; Blakeslee et al. 2009). The ESO archive contains a single MUSE pointing of the galaxy, with a footprint as shown in Figure 28 (ESO Archive ID: ADP.2018-03-26T15:02:26.469, PI: M. Sarzi, Program ID: 296.B-5054). This MUSE-DEEP data cube was assembled from two observations, with an effective observation time of 4921 s, and a seeing at 5007 Å of 087. Owing to the low luminosity of the host galaxy (the MUSE data cube encompasses only V ∼ 12.9 of galaxy light), just 21 PNe were detected, with only ∼12 in the top ∼1 mag of the luminosity function where the data are likely complete. This limits our ability to constrain the system's distance.

Figure 29 shows the cumulative PNLF of the galaxy. As with NGC 1366, the shape of the bright end of the distribution is in good agreement with the empirical law defined by Equation (2). However, the exact magnitude of the PNLF cutoff is not well defined due to the limited PN data set.

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We measured the amount of galaxy light underlying the position of each PNe and over the MUSE data cube as a whole using the B-band surface photometry of de Carvalho et al. (1991); these data were then converted to V by assuming a color of (B − V) = 0.89 (Prugniel & Heraudeau 1998). To estimate the line-of-sight velocity dispersion, we used the measurements of Graham et al. (1998) for objects in the inner ∼20'' of the galaxy and adopted 50 km s⁻¹ for PNe at larger galactocentric radii.

The results of our analysis are shown in Figure 30. With a reddening of E(B − V) = 0.011, our fit to the PNLF gives ${(m-M)}_{0}={31.39}_{-0.26}^{+0.10};$ when we include a ∼0.06 mag error due to the flux calibration, aperture correction, and extinction our final value is ${(m-M)}_{0}={31.39}_{-0.27}^{+0.12}$, or ${18.9}_{-2.5}^{+1.1}$ Mpc. This value is consistent with membership in the Fornax cluster and is essentially identical to our distance to NGC 1404, but on the low side of the distribution of SBF distances and smaller than the most recent SBF distance of d = 22.9 ± 0.9 Mpc (Blakeslee et al. 2009). Given the known zero-point offset between the PNLF and SBF distance scales, the difference is a bit larger than expected.

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NGC 1433 is a barred spiral galaxy (Hubble type (R')SB(r)ab), which shows a double ring structure, an AGN, and a bright central region of intense star formation. The galaxy has been observed extensively as part of the PHANGS program (PI: E. Schinnerer, Program ID: 1100.B-0651), from which we have used their nine MUSE pointings. The IDs of these data cubes, along with their exposure times and seeing at 5007 Å are listed in Table 5; their positions are shown in Figure 31.

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Table 5. Data Cubes for NGC 1433

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
P1	ADP.2019-12-22T01:43:14.831	2580	088
P2	ADP.2019-12-22T01:22:27.745	2580	090
P3	ADP.2019-12-17T22:49:20.671	2580	094
P4	ADP.2019-11-27T01:02:00.695	2580	093
P5	ADP.2017-06-14T09:12:09.412	3840	089
P6	ADP.2019-11-27T02:44:54.831	2580	073
P7	ADP.2020-01-09T22:41:58.291	2580	074
P8	ADP.2020-01-11T02:39:27.620	2580	076
P9	ADP.2020-01-08T07:08:40.807	2580	059

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The identification and measurement of PNe in NGC 1433's were made difficult by two problems: diffuse emission from the system's interstellar gas, and the lack of point sources (i.e., PSF stars) in the field. The former issue increases the random photometric errors of the PN measurements, as the complex morphology and excitation of the emission-line gas results in uncertain background subtractions around the [O iii] λ5007 and Hα lines. The latter problem introduces a systematic error into the photometry of each pointing, as without bright PSF stars, it is impossible to accurately determine the aperture correction applicable to our faint-object photometry. When the data from all nine fields are combined, this systematic error then manifests itself as a pseudo-random error, which smooths out and distorts the shape of the PNLF. The net effect of this smoothing is to bias the PNLF results toward smaller distances.

Using the DELF technique we identified almost 500 pointlike [O iii] λ5007 sources across NGC 1433's disk; after examining the spectra of these objects, almost half were excluded as interlopers. This left us with 258 PN candidates, with over 120 in the top ∼1 mag of the luminosity function. As the lower panel of Figure 32 illustrates, the luminosity function of these data is very well defined and complete to at least m₅₀₀₇ = 28.2. However, the bright-end cutoff of the PNLF has a shape that is closer to that of a power law than an exponential. Part of this behavior is due to the existence of two sources that are ∼0.3 mag more luminous than the next brightest planetary. A careful inspection of the DELF-extracted data cube surrounding these sources reveals that their [O iii] measurements are likely contaminated by light from the regions' bright and irregularly distributed emission-line gas. However, even if these two bright objects are excluded, the shape of the PNLF's cutoff is still somewhat less abrupt than expected.

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The behavior of NGC 1433's PNLF is consistent with the hypothesis that the PN photometry of the galaxy is compromised by the zero-point issues described above. The system's nine non-overlapping data cubes were taken under a range of seeing conditions (from 064–105 seeing), which made placing all the PN candidates on a unified flux scale difficult. This is evidenced by the distribution of PN magnitudes within the fields. Pointing P5 contains no good PSF stars, and the zero-point error associated with the uncertain aperture correction may be as large as 0.1 mag. Not coincidentally the field contains the three brightest PNe in our sample. Pointing P3, which also has a highly uncertain (σ ∼ 0.1 mag) zero-point due to its lack of PSF stars, contains the fourth and fifth brightest PNe. Conversely, the most luminous PNe in P2, a third field with no bright point sources, is 0.3 mag fainter than the brightest PN in P6, a pointing that does have a reliable aperture correction.

Thus, it is quite likely that the bright tail of NGC 1433's PNLF is entirely due to zero-point errors in the PN photometry. If so, there are three ways to handle this type of error. The first is simply to discard the PNe in fields where the aperture correction and/or flux calibration is untrustworthy. This removes a third of the sample, but, as shown in the upper panel of Figure 32, it produces a PNLF more in line with the expectations of Equation (2).

A second possibility is to artificially shift the PNLF's of those MUSE pointings with poor photometric zero-points to match the data acquired from data cubes with robust aperture correction measurements. If the PN samples within the data cubes were larger, this might be a viable methodology. However, since each cube only contains five or six PNe in the top magnitude of the luminosity function, it would be impossible to determine the appropriate shifts with the accuracy needed for coaddition.

The third way to handle variations in each data cube's zero-point is to increase the random photometric errors assigned to each individual PN. This, of course, is an approximation and somewhat arbitrary: the aperture correction errors of each data cube act in a systematic fashion, shifting all the PN magnitudes in the same manner. However, if one assumes that (a) each MUSE pointing contains a similar number of PNe and (b) the distribution of zero-point errors over the nine data cubes is roughly Gaussian, then the expected distortion in the PNLF's shape can be easily modeled and fit to the data in the manner outlined by Chase et al. (2023). For example, the curves shown in the bottom panel of Figure 32 all assume an apparent distance modulus of (m − M) = 31.35 and the PNLF of Equation (2). However, while the black curve shows the expected PNLF in the absence of additional errors, the red and green curves show what the curve would look like with an additional 0.1 and 0.2 mag of randomly distributed errors, respectively. These errors, which could either be due to poor PN background subtraction or the effect of imperfect aperture corrections in each of the nine fields, produce a function that adequately represents the observed data.

To derive the best-fit curves shown in Figure 32 we started with the galaxy continuum measurements from the MUSE data cube. While Buta (1986) and Buta et al. (2001) do provide a plot of NGC 1433's azimuthally averaged surface brightness profile, the galaxy's complex two-dimensional morphology, with its bulge, bar, and ring-like structures precludes the use of any simple photometric model. Consequently, we estimated the amount of light underlying each PN directly from stellar continuum measurements on the MUSE data cubes. Excluding the central ∼3'' of the galaxy, where PN detections are difficult, the total amount of V-band light contained in the nine MUSE pointings is V ∼ 10.5. We also adopted a single number for the galaxy's line-of-sight stellar velocity dispersion. Buta et al. (2001) have shown that in the inner ∼30'' of the galaxy, this number is ∼75 km s⁻¹, a value significantly smaller than the velocity difference needed to deblend the emission lines of two superposed PNe. Consequently, we simply use this number throughout the galaxy.

Figure 33 uses the entire nine field data sample to illustrate how the inferred distance to NGC 1433 depends on the accuracy of the photometry. If the formal errors of the photometry are accurate, then the galaxy's most likely apparent distance modulus is ${31.21}_{-0.04}^{+0.04}$, though there is some likelihood that superpositions are biasing the result. However, if we assume that there are additional sources of photometric error, such as those produced by incorrect estimates of the PSF aperture corrections or poor sky subtraction of the galaxy's diffuse emission, then the best-fit distances increase to $(m-M)={31.35}_{-0.06}^{+0.04}$ (for σ_add = 0.1 mag) and $(m-M)={31.49}_{-0.07}^{+0.06}$ (for σ_add = 0.2 mag).

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To understand the trend seen in Figure 33, consider the basis behind the PNLF method. PNLF distances are defined by measuring the shape of the function's rapid bright-end cutoff. In defining this shape, a single PN near M* carries more weight than a similar PN at fainter magnitudes—the brighter the PN, the greater its effect on the measured value of M*. Consequently, random photometric errors will turn into a systematic fitting error, and an underestimate of galaxy distance. The only way to avoid this systematic is to carefully model the effect of errors on the PNLF, either through the application of an Eddington (1913) correction to the binned data, or by fitting the PN magnitudes, not to the expression given by Equation (2), but to the result of a convolution of the fitted function with a kernel that represents the expected amplitude of the errors (Ciardullo et al. 1989).

In the case of NGC 1433, the aperture corrections for pointings P2, P3, and P5 are poorly known, as are their errors. In the absence of this information, magnitude excursions to the bright side of the true value push the resultant distance to smaller values more than those excursions that work the other way. Thus, the PNLF's best-fit distance modulus is underestimated.

A confirmation of this effect comes from the PNLF solution when objects from the three fields with poorly determined aperture corrections, P2, P3, and P5, are removed from the analysis. For these higher-quality data, the distance modulus for an assumed E(B − V) = 0.008 is ${(m-M)}_{0}={31.42}_{-0.06}^{+0.04}$. If we then assume a nominal uncertainty of 0.06 mag for the flux calibration and aperture correction, then the distance to the galaxy becomes ${(m-M)}_{0}={31.42}_{-0.08}^{+0.07}$, or 19.2 ± 0.7 Mpc.

Our result for NGC 1433 is essentially identical to that of Scheuermann et al. (2022), who use the same MUSE data cube to identify 90 PNe in the galaxy and derive a distance modulus of ${(m-M)}_{0}={31.39}_{-0.07}^{+0.04}$ (${18.94}_{-0.46}^{+0.39}$) Mpc. We support their explanation of the conflict between our PNLF distance and the TRGB result of ∼9 Mpc (Sabbi et al. 2018): confusion between RGB and AGB stars. The PNLF of Figure 32 rules out a distance modulus that would be smaller by as much as 1.6 mag: PNe that bright are simply not seen in this galaxy.

NGC 1512 is a barred spiral galaxy, classified SB(r)a, with both a traditional outer ring and an inner disk that encompasses a UV-bright nucleus with intense star formation. As the galaxy is relatively inclined (i ∼ 72°), it has been the target of a myriad of Tully–Fisher investigations, all of which place the system between ∼11 and 15 Mpc away (see Tully et al. 2009). A TRGB analysis derived from multiwavelength HST photometry agrees with this assessment, assigning a distance at the near end of this range (11.6 Mpc; Sabbi et al. 2018). A grid of nine MUSE pointings obtained by the PHANGS program (PI: E. Schinnerer, C.M. Carollo, IDs: 1100.B-0651, 099.B-0242) covers the galaxy and is available in the ESO archive. These pointings are listed in Table 6 and shown in Figure 34.

Table 6. Data Cubes for NGC 1512

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
P1	ADP.2019-02-05T06:15:31.835	2580	082
P2	ADP.2018-03-02T17:54:29.654	3225	175
P3	ADP.2018-03-08T17:26:28.765	2580	073
P4	ADP.2019-02-05T06:15:31.827	2580	081
P5	ADP.2017-12-12T10:52:12.342	3600	082
P6	ADP.2018-03-08T18:22:48.347	2580	076
P7	ADP.2019-02-12T01:24:29.983	2580	127
P8	ADP.2019-02-12T01:24:30.018	2580	110
P9	ADP.2019-02-12T01:24:29.997	2580	089

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Since there is little to no surface photometry available for NGC 1512, we used continuum measurements from the MUSE IFUs to estimate the amount of galaxy light falling onto the nine data cubes and at the position of each PN. Also, since NGC 1512 is a disk galaxy, we did not attempt to model the system's line-of-sight velocity dispersion, as we expect this number to be much less than the minimum velocity separation that can be deblended by the MUSE spectrograph. In our analysis, we adopted 50 km s⁻¹ for this value.

PN detections within an isophotal radius of ∼13'' of NGC 1512's nucleus are extremely difficult, due to the region's high surface brightness and bright diffuse emission, so we eliminated the region from our analysis. Also, poor seeing precluded faint point-source detections in fields P2 and P7, so these areas were also excluded from consideration. That left us with V ∼ 11 mag of galactic light hosting 210 PN candidates found by our DELF analysis. Over 60 of these objects are in the top ∼1 mag of the luminosity function. Figure 35 displays the object's PNLF.

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As Figure 35 illustrates, the bright end of NGC 1512's PNLF is very well defined, with a cutoff magnitude of m₅₀₀₇ ∼ 27. However, there is one slight ambiguity: the shape of the PNLF, as defined by the PNe with m₅₀₀₇ > 27, suggests a PNLF cutoff that is very close to m* ∼ 26.9. Yet the brightest PN in the galaxy (PN1) is ∼0.1 mag brighter than this threshold. In other words, the object appears slightly overluminous compared to the apparent PNLF cutoff. One potential explanation for this brightness is a PN superposition.

This possibility is summarized in Figure 36. Because the region upon which PN1 is superposed is relatively bright (μ_V ∼ 21), the likelihood that the object is composed of two superposed sources is moderately high. But still, the hypothesis is disfavored: the best-fit solution considers PN1 to be a single object and yields a distance modulus of ${(m-M)}_{0}={31.30}_{-0.04}^{+0.04}$, or 18.2 ± 0.3 Mpc (for an E(B − V) = 0.01). Yet if PN1 is a single, ordinary PN, the quality of the fit is rather poor, and a K-S test can rule out the null hypothesis with 86% confidence. Conversely, if PN1 is a superposition of two objects, the preferred solution is (m − M)₀=31.43 ± 0.04; not coincidentally, this is identical to the distance of ${31.43}_{-0.04}^{+0.03}$ (${19.3}_{-0.4}^{+0.3}$ Mpc) which is derived when PN1 is excluded entirely from the sample. In this case, the fit to Equation (2) is excellent and the K-S statistic is very small at D_n = 0.062. We therefore prefer the latter value for the distance.

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Like NGC 1433, the PNLF of NGC 1512 is compromised by uncertainties in the zero-point errors of some of the data cubes. Specifically, pointings P3, P6, and P8 show an unusual wave-like pattern in the aperture correction versus wavelength relation and/or have noise estimates which may point to an issue in the data reduction process. Fortunately, unlike NGC 1433, the PNe of these data cubes do not dominate the bright end of the luminosity function: the second and sixth brightest PNe are in P6, while P3 and P8 contribute very little to the definition of the PNLF's cutoff. Since we estimate that the flux calibration and aperture correction errors for the remaining fields are no greater than 0.05 mag, we adopt this number as the systematic component of our error budget. Thus, if we assume a foreground reddening of E(B − V) = 0.01, our final distance to the galaxy is ${(m-M)}_{0}={31.43}_{-0.06}^{+0.06}$ (${19.3}_{-0.6}^{+0.5}$ Mpc).

Our result is slightly larger than the distance modulus of ${(m-M)}_{0}={31.27}_{-0.11}^{+0.07}$ found by Scheuermann et al. (2022) from the same data cube, albeit with a much smaller sample (43) of PNe. However, it is significantly larger than the value of ∼11.7 Mpc found from the analysis of UV and optical HST images taken as part of the Legacy Extragalactic UV Survey (Sabbi et al. 2018). As noted for NGC 1433, the TRGB distance is much smaller than both PNLF distance estimates and probably for the same reason discussed above, i.e., confusion between the TRGB and that of the AGB.

NGC 2207 and IC 2163 are a pair of interacting spiral galaxies in the early stages of a merger. The larger system, NGC 2207, is classified SAB(rs)bc pec and has a total B magnitude of 10.8; IC 2163 is a barred spiral of type SB(rs)c pec with B = 11.4 (de Vaucouleurs et al. 1991). Both galaxies exhibit robust star formation and are subject to a relatively large amount of foreground extinction (A_V = 0.238; Schlafly & Finkbeiner 2011).

There are no TRGB or Cepheid distances to the system, and estimates from the Tully–Fisher relation (13–17 Mpc; Bottinelli et al. 1984, 1986; Russell 2002; Theureau et al. 2007) are discordant with that determined from SN Ia 1975A (33–50 Mpc; Arnett 1982; Davis et al. 2021). Thus, a PNLF distance to the galaxies would be interesting. However, if the SN-based distances are correct, then it would be a real challenge to measure a PNLF solely from the MUSE archival data.

Three MUSE data cubes of the NGC 2207/IC 2163 system exist in the ESO archive; these data are summarized in Table 7. Field P1, which is centered on the nucleus of NGC 2207 was surveyed as part of Program 0102.D-0095 (PI: J. Anderson); data for field P3 (which covers most of IC 2163) and P2 (which connects P1 and P3) were obtained through Program ID: 0100.B-0116 (PI: C.M. Carollo). As can be appreciated from the difference image in Figure 37, the system's copious diffuse emission, coupled with the data set's relatively short exposure times and only moderate image quality, made PN detections challenging. Nevertheless, the two galaxies are luminous enough so that, if the system were at the distance implied by the Tully–Fisher measurements, then a large number of PN candidates would be observable.

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Table 7. Data Cubes for NGC 2207/IC 2163

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
P1	ADP.2019-03-06T02:54:12.126	2238	088
P2	ADP.2018-04-05T08:26:13.265	3338	066
P3	ADP.2018-04-05T08:26:13.281	2892	084

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Our initial examination of the MUSE data cubes identified 112 pointlike [O iii] sources in the two galaxies. However, after screening the objects' spectra for evidence of strong Hα (typical of an H ii region) and emission from [S ii] and [N ii] (the signature of an SNR), only three of the PN candidates survived. These potentially true PNe are all near the detection limit of the survey and have [O iii] magnitudes of m₅₀₀₇ = 28.77 ± 0.13, 29.70 ± 0.28, and 29.91 ± 0.33.

If the brightest of the objects is indeed a PN with an absolute magnitude near M*, then the implied distance to the galaxy is close to ∼40 Mpc, in agreement with the analysis of SN 1975A. Scaling from Paper I's survey of NGC 474, an elliptical galaxy at a similar distance, a proper PNLF study of the system would likely require at least ∼5 hr of MUSE exposure time with ∼06 seeing. With the present data, all we can say is that the PN observations are inconsistent with the galaxies' Tully–Fisher distances.

NGC 3501 is an edge-on spiral, with an uncertain Hubble type of Scd. The galaxy has more than a dozen Tully–Fisher distance estimates in the literature, all within the range of 19.5–26.5 Mpc. However, the viewing angle presents an obvious challenge to standard candles such as Cepheids and the PNLF, and no TRGB distances have been published. Nevertheless, it has been shown that [O iii] observations in the halos of edge-on systems can overcome the issues associated with galaxy orientation and produce reliable PNLF distances (e.g., Ciardullo et al. 1991; Jacoby et al. 1996). So MUSE has the potential to obtain a PNLF distance to the galaxy.

As depicted in Figure 38, the one NGC 3501 MUSE data cube in the ESO archive (ID: ADP.2017-10-16T11:12:01.527, PI: F. Pinna, Program ID: 098.B-0662) consists of the combination of two pointings. The effective exposure time for these data is quoted as 9600 s, but the image quality of the observation is quite poor, 151. Nevertheless, the DELF technique did allow us to detect six of the system's brightest PNe, from an initial list of 17 candidates. Such a sparse sample is not suitable for a PNLF analysis, though it can be used to place limits on a galaxy's distance.

It is worth noting that, although the magnitudes of the detected PNe range from 27.69 ≤ m₅₀₀₇ ≤ 28.47, the difference between the brightest and second brightest PN is 0.54 mag. This gap may simply be due to the sparsity of the sample. However, a careful examination of the [O iii] line of the brightest PN hints at the possibility that the object is actually composed of a marginally resolved pair of sources; if so, the two objects are of comparable brightness and have a separation of ∼04 with a position angle of 125° (Figure 39). Unfortunately, the poor image quality does not allow us to draw any definite conclusions about the source.

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As with NGC 2207, we can estimate a crude upper limit to NGC 3501's distance using the magnitudes of the brightest PNe. If we believe that the brightest [O iii] source is a superposition of two objects and adopt the second brightest object as our indicator, then, for a foreground reddening of E(B − V) = 0.02, the upper limit on the galaxy's distance is ∼38 Mpc. Alternatively, if we believe the [O iii] flux from PN1 comes from a single PN, then the upper limit drops to ∼33 Mpc. Either way, these estimates are well beyond the system's Tully–Fisher values. We emphasize that this estimate is for an upper limit on the distance, so it is still consistent with other measurements. The only firm conclusion we can draw is that, under more favorable observing conditions, MUSE should be able to yield an accurate PNLF distance to the galaxy.

The Antennae galaxy (NGC 4038/NGC 4039) is the closest, and therefore the prototypical, example of a major merger, and is well known for its spectacular antennae-like tidal tails (hence its name) and vigorous starburst activity. A PNLF study of the Antennae is particularly interesting, both because of an earlier controversial distance determination for SN 2007sr (Schweizer et al. 2008), and because it has been the focus of both TRGB and Cepheid observations. In principle, NGC 4038/39 can serve as a checkpoint for four different distance indicators, although the PNLF and SN Ia methods have ties to the Cepheid and/or TRGB scales and thus, are not fully independent.

NGC 4038/39 has been observed extensively with MUSE to study the ionization mechanism of the DIG (Weilbacher et al. 2018). As a result, there are numerous data cubes available in the ESO Archive (PI: P. Weilbacher, Program IDs: 093.B-0023, 095.B-0042). However, we chose to optimally re-reduce all the cubes with the most recent version of the MUSE pipeline (Weilbacher et al. 2020); this process removed many of the artifacts (such as Hα saturation and high sky-line residuals) present in the prior reduction. The re-reduction also allowed us to produce a set of 13 independent data cubes without the complication of changing aperture corrections in the regions of field overlap. These fields are listed in Table 8 and illustrated in Figure 40.

Table 8. Data Cubes for NGC 4038/39

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
C01	ADP.2017-03-28T13:08:20.713	4951	074
C02	ADP.2017-03-28T13:08:20.697	5013	059
C03	ADP.2017-03-28T13:08:20.689	4811	062
C04	ADP.2017-03-28T13:08:20.681	4985	059
C05	ADP.2017-03-28T13:08:20.705	5117	084
C06x	ADP.2016-06-15T08:55:13.262	2578	135
C09	ADP.2017-05-24T12:39:01.277	2584	081
C10	ADP.2016-09-29T05:21:54.086	2592	086
C11a	ADP.2017-05-24T11:10:28.457	2498	098
C11b	N/A a	2523	078
C12a	N/A a	2700	088
C12c	N/A a	2700	075
C12e	N/A a	2700	060

Note.

^a Not available in a public archive, own data reduction (see the text).

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Our DELF analysis found 234 PN candidates scattered over the 13 fields, with almost a hundred in the top ∼1 mag of the luminosity function. In general, the spatial distribution of these PNe does not follow that of the light: in the regions of intense star formation and high obscuration, PN detections are difficult, and little effort was made to perform a comprehensive search in those areas. However, the large number of PNe found between and beyond the regions of star formation creates a very well-defined PNLF. This function is shown in Figure 41.

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As is illustrated in the figure, the distribution of observed PN magnitudes in the Antennae galaxy is generally well fit by the empirical luminosity function described by Equation (2). But there is one exception: PN1, which is 0.14 mag more luminous than the second and third brightest objects, appears slightly too bright for the best-fit curve. Moreover, it is difficult to argue that the PN's [O iii] flux is due to a superposition of two sources, as the location upon which the object is projected is not especially bright. Specifically, in terms of underlying surface brightness, the position of PN1 barely falls in the top one-third of the sample. But its existence forces the best-fit curve toward smaller distances.

The effect of PN1 is quantified in Figure 42, where we fit Equation (2) to the observed distribution of PN magnitudes with and without PN1. For the analysis, we assumed that the line-of-sight velocity dispersion of the galaxy's stars is ∼150 km s⁻¹ throughout the MUSE survey region. Given the complexity of the system and the lack of any spatially resolved stellar radial velocity measurements, this was the simplest assumption we could make. Similarly, due to the irregular isophotes of the galaxy, our estimates for the amount of galaxy luminosity underlying the position of each PN were made from the MUSE spectra themselves, rather than any published surface photometry. These data imply that, after excluding those regions of the galaxy where PN detections were impossible, the 13 pointings shown in Figure 40 contain V ∼ 10.8 mag of galaxy light.

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Figure 42 confirms that, due to the large number of PNe in the top ∼1 mag of the luminosity function, the formal errors of the PNLF fits are quite small. If PN1 is excluded, the distance modulus of the system (for an E(B − V) = 0.04) is ${(m-M)}_{0}\,=\,{31.86}_{-0.04}^{+0.03}$, or 23.6 ± 0.4 Mpc. In other words, the statistical error for the distance is less than 2%. However, if PN1 is included, then the Antennae galaxy's most likely distance modulus decreases by 0.048 mag, a value that is greater than the statistical error of the fit. The fit with PN1 is slightly worse than that without the object, but not so much that it can be deemed inconsistent with the shape of Equation (2). Since both solutions are easily allowed by the K-S statistic, we prefer keeping PN1 in the data set. The distance is then ${(m-M)}_{0}\,=\,{31.82}_{-0.04}^{+0.03}$, or 23.1 ± 0.4 Mpc, where the uncertainties represent only the internal errors of the fits.

Since the Antennae system was observed using 13, sometimes overlapping MUSE pointings, the analysis of the systematic component of the distance determination's error budget is complicated. Most of the PNe that occupy the brightest 0.25 mag of the PNLF lie in fields C10, C5, C9, and C1. None of these fields have any anomalies in the aperture correction estimates: formally, the uncertainties in their aperture corrections are 0.029, 0.010, 0.045, and 0.001 mag, respectively. Conversely, the C4 pointing contains no suitable pointlike object for an aperture correction measurement; the best we could do is stack the field's brightest PNe and determine the aperture correction for [O iii] λ5007 as explained in Paper I. This procedure likely has an uncertainty of ∼0.05 mag. For pointing C11b, which also contains no useful PSF stars, we took a different tack: since the seeing of the data cube was extremely similar to that of pointing C1, we simply used the latter's aperture correction in our analysis. Fortunately, the PNe contained in the pointings with the poorest aperture correction measurements do not contribute significantly to the PNLF's bright-end cutoff. We, therefore, adopt ∼0.06 mag as the combined zero-point uncertainty due to the MUSE data cube's flux calibration and aperture correction.

Our final distance to NGC 4038/39 is ${(m-M)}_{0}\,=\,{31.82}_{-0.07}^{+0.07}$ (or ${23.1}_{-0.8}^{+0.7}$ Mpc) for an E(B − V) = 0.04. This distance is statistically identical to the SN Ia-based value of (m − M)₀ = 31.74 ± 0.27 (Schweizer et al. 2008), and is only slightly larger (but still consistent with) the system's TRGB distances, which fall near 31.67 ± 0.05 (Jang & Lee 2015; Freedman et al. 2019). Compared to the system's Cepheid distances, our value is much larger than the discordant estimate of 31.29 ± 0.11 by Riess et al. (2016) but is in reasonable agreement with the measurements of (m − M)₀ = 31.55 ± 0.06 (Fiorentino et al. 2013) and 31.615 ± 0.117 (Riess et al. 2022). These results suggest that PN1 does belong in the sample, and the 0.14 mag offset between PN1 and PN2 is merely a statistical fluke, possibly due to the zero-point differences between the different MUSE pointings.

One final issue to keep in mind is that some of the dust associated with the NGC 4038/39's starburst regions may have been forced outward over the body of the galaxy by the tidal forces associated with the interaction. In such a scenario, this dust could extinct some of the PNe and cause an overestimate of the galaxy's distance. The fact that our PNLF distance to the system is slightly larger than those produced by the TRGB and Cepheid techniques lends support for this internal extinction hypothesis.

The giant E3 elliptical galaxy NGC 4365 is the central member of Virgo's ${{\rm{W}}}^{{\prime} }$ group, roughly 6 Mpc behind the system's central core (Blom et al. 2014). The galaxy is a popular target for studies of extragalactic globular clusters, and it has been analyzed kinematically with the aid of the Planetary Nebula Spectrograph (Pulsoni et al. 2018). The published SBF distances to the galaxy span the range of 16.9 < D < 24.4 Mpc, with the latest SBF measurement placing the galaxy at 23.1 Mpc (Blakeslee et al. 2009).

There is a single MUSE data cube of NGC 4365 in the ESO archive (ID: ADP.2016-06-07T11:11:26.095, PI: L. Coccato, Program ID: 094.B-0225). The exposure, which is centered on the galaxy's nucleus (see Figure 43), has an exposure time of 2343 s, and an image quality of 076 at [O iii] λ5007.

Our DELF analysis initially found 115 pointlike [O iii] sources in the MUSE data cube; however, upon closer inspection of their spectra, 51 were excluded as either spurious objects (i.e., too faint to meet the detection criteria) or some type of interloper. That left us with 64 PN candidates, with ∼50 in the top ∼1 mag of the PNLF.

As illustrated in Figure 43, only five of these detections were made in the galaxy's central ∼12'', an area that contains ∼35% of the light falling on the MUSE frame. This is not unexpected, as faint-object detections are severely incomplete against a bright, rapidly varying background. We therefore eliminate the region and its PNe from the analysis.

Interestingly, one of the excluded objects is PN1, a source that is 0.37 mag brighter than any other PN candidate in the galaxy. PN1 is located just 51 from NGC 4365's nucleus, on an isophote with an R-band surface brightness of 17.2 mag arcsec⁻² (Lauer 1985). If we assume a color of (V − R) = 0.60 (Buta & Williams 1995) and a distance of 24 Mpc (Blakeslee et al. 2009), then Equations 3, 4, and 5 of Chase et al. (2023) imply that there is a ∼30% chance that the observed [O iii] flux is formed from the emission of multiple sources. In fact, a careful inspection of the narrowband images made from the MUSE data cube reveals that the position of PN1 shifts slightly (by 034) between the wavelengths of 5021.5 and 5024.0 Å. This supports the hypothesis that the [O iii] flux of PN1 is produced by multiple superposed sources. Thus, the extreme luminosity of the source is not surprising and further justifies its rejection in the PNLF analysis.

The luminosity function of PNe outside the central 12'' of NGC 4365 is shown in Figure 44. The function is very well defined and shows no obvious deviation from the empirical law of Equation (2): the PN number counts decline rapidly to zero at magnitudes brighter than m₅₀₀₇ ∼ 27.2, and there is no evidence for the existence of any overluminous objects. Although the data are only complete over the brightest ∼0.8 mag of the luminosity function, the data produce a very precise measure of distance.

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To fit the PN magnitudes of Figure 44, we used the surface photometry of Lauer (1985) and Michard & Simien (1988), and the kinematic data of Foster et al. (2016) to estimate the amount of galaxy light and the line-of-sight velocity dispersion at every location in the data cube. The former measurements imply that the MUSE survey contains V ∼ 10.8 of galaxy light; the latter quantities range from ∼225–207 km s⁻¹. (Both values exclude the central 12'' of the galaxy.) We then applied the algorithms of Chase et al. (2023) to derive the likelihood contours of Figure 45. Assuming a foreground reddening of E(B − V) = 0.02, the resultant distance modulus to the system is ${(m-M)}_{0}\,=\,{31.55}_{-0.08}^{+0.05}$.

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The systematic component of NGC 4365's error budget is quite low, as the data cube contains a bright and well-behaved PSF star. (The formal error on the aperture correction is just 0.002 mag.) If we assume a nominal 3% error due to the MUSE data cube's flux calibration, then the distance modulus to the galaxy becomes ${(m-M)}_{0}\,=\,{31.55}_{-0.09}^{+0.06}$ or ${20.4}_{-0.8}^{+0.6}$ Mpc. This value is on the low end of the range of SBF results, but consistent with the known zero-point offset between the two distance scales, which can be as high as 15% (Ciardullo et al. 2002).

NGC 4418 (also known as NGC 4355 and IRAS 12243-0036) is a lenticular galaxy (Hubble type S0/a), with unusually luminous IR emission and mid-IR silicate absorption features that indicate a bright, but deeply obscured nucleus. The nature of the nuclear region (starburst or AGN) is unclear, but evidence from radio data suggests the presence of a compact super-star cluster with intense star formation (Varenius et al. 2014). The only distances available to the system come from a single Tully–Fisher analysis, which places the galaxy ∼22 Mpc away (Theureau et al. 2007).

One promising MUSE-DEEP data cube exists in the ESO archive (archive ID: ADP.2020-02-13T10:53:27.677, PI: F. Stanley, Program ID: 0104.B-0668); these data have an exposure time of 5999 s, an image quality of 089 at 5007 Å, and a position centered on the galaxy's nucleus (see Figure 46). Our examination of the DELF images created from the data cube resulted in a sample of 56 PN candidates, of which 47 survived spectral examination.

The locations of NGC 4418's PN candidates are shown in Figure 46. Within the 11'' semimajor axis isophote, there are no PN candidates; this reflects the effect that the region's high surface brightness has no PN detections. The PNLF of the objects outside this radius is shown in Figure 47.

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NGC 4418's PNLF is not as well populated as some of the other galaxies in this study. The data start to become incomplete ∼0.7 mag down the PNLF, and there are only ∼24 PN detections brighter than this limit. Consequently, stochasticity is an issue. For example, PN1 is 0.27 mag brighter than any other planetary. In galaxies with a well-populated PNLF, such a gap might be indicative of an anomaly. Here, it is perfectly consistent with expectations. In fact, a K-S test prefers a fit that includes PN1.

NGC 4418 has no published surface photometry, nor are there any measurements of the disk's line-of-sight velocity dispersion. We compute the former using continuum measurements from the MUSE spectra; after excluding the galaxy's inner ∼11'', we estimate that V ∼ 13.6 of light is contained in the survey region. For the latter we adopt 75 km s⁻¹ throughout the galaxy; this number is much less than the minimum velocity separation needed to deblend two sources and is consistent with expectations for the velocity ellipsoid of a disk galaxy.

Figure 48 displays the likelihood contours for NGC 4418. The distortion of the lowest-level contours reflects the possibility that PN1 may be a superposition: the galaxy continuum underlying the PN is relatively bright and the object is the most luminous PN in the sample. However, as the marginalized probability curve in the lower panel of the figure indicates, the possibility has almost no effect on the most likely distance to the galaxy: for a foreground reddening E(B − V) = 0.02 (Schlafly & Finkbeiner 2011) we obtain ${(m-M)}_{0}\,=\,{32.59}_{-0.10}^{+0.07}$. When we include the small systematic component to the error budget (as shown in Figure 46, the error on the aperture correction error is minimal due to the presence of a bright field star), the distance modulus becomes ${(m-M)}_{0}\,=\,{32.59}_{-0.10}^{+0.07}$, or ${32.9}_{-1.5}^{+1.2}$ Mpc. This places the galaxy among the most distant objects ever studied by the PNLF technique, and well beyond the galaxy's Tully–Fisher estimate. The observation demonstrates what is now possible with the MUSE instrument.

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We do note that, since the galaxy is a bright IR source, it is possible that some of the emitting dust could be distributed across the face of the galaxy. So some of the PNe may be attenuated by this component. In that case, our PNLF distance would be an overestimate. Assuming that this is not the case, the galaxy provides an opportunity for direct measurement of the Hubble constant, as the system is distant, relatively isolated, and has an unusually bright field star projected onto its body for a precise measure of the aperture correction (see Section 6.4).

The E2 giant elliptical NGC 4472 is the brightest galaxy in the Virgo cluster and is at the center of the Virgo B subclump. Previous PNLF measurements have placed this galaxy at a distance of 13.9 Mpc (Jacoby et al. 1990), 14.4 Mpc with a new calibration from the HST key project (Ferrarese et al. 2000), and 18.1 Mpc (Hartke et al. 2017). NED lists 22 SBF distances for the system, ranging from 14.3–17.8 Mpc.

There are a total of nine relatively shallow MUSE data cubes of NGC 4472 in the ESO archive (Program ID: 095.B-0295 PI: J. Walcher) that are provided as three mosaics; each combines three MUSE pointings in one cube. We list these data in Table 9 and show their locations in Figure 49. Despite the short exposure times, we managed to identify and measure 81 of the galaxy's PN candidates with the DELF technique.

Table 9. Data Cubes for NGC 4472

Field	Archive ID	Exp Time	Seeing
		(s)	(5007 Å)
M1	ADP.2016-06-21T00:50:22.928	626	072
M2	ADP.2016-07-11T13:38:27.884	623	082
M3	ADP.2016-06-21T00:31:05.332	621	093

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Figure 50 displays the PNLF of the objects. In the figure, we have excluded the PNe in the central ∼40'' of the galaxy, since, as Figure 49 illustrates, the limited depth of the data made PN detections in this high surface brightness region difficult. Even outside this radius, the PN luminosity function is not especially deep, as the data only extend ∼0.7 mag fainter than PNLF's bright-end cutoff. This is just deep enough to determine a robust distance to the galaxy.

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To determine the amount of light underlying the location of each PN, we used the surface photometry of Cohen (1986); outside the 40'' radius, the total amount of galaxy light falling onto the MUSE data cubes is V ∼ 9.5 mag. The stellar velocity dispersion, as a function of galactic radius, was taken from Veale et al. (2017). For the PNe being analyzed here, this quantity ranges from ∼230 to ∼255 km s⁻¹. Effectively, this means that even if two PNe are spatially colocated on the same location on the sky, there is better than a 50% chance that the flux from the two objects can be disentangled via their differing radial velocities. This is consistent with a visual inspection of Figure 50: our PNLF displays no evidence for any superposed or overluminous objects.

The aperture correction error for this data set is particularly difficult to assess. The $3^{\prime} \times 3^{\prime}$ region surveyed by MUSE is composed of three mosaics, and each mosaic consists of data from three MUSE pointings. Consequently, assumptions must be made as to whether the aperture correction derived from any given star is applicable to the faint sources found in other regions of the same mosaic. Moreover, the short exposure time means that the data are rather shallow, leading to low S/N measurements. Mosaic M1 does contain one very bright star that yields a formal aperture correction error of 0.001 mag, and the photometry of two other stars in the field yields corrections that are consistent with that from the bright star. At least in the M1 mosaic, the zero-point error appears to be dominated by the uncertainty of the MUSE flux calibration, rather than our ability to define the point-source aperture correction.

In contrast, M2 covers the high surface brightness region around NGC 4472's nucleus, where faint pointlike sources are difficult to measure. In this mosaic, the formal aperture correction errors may be as large as 0.17 mag. M3 lies somewhere in the middle: our value for the aperture correction seems to be reasonably accurate, although the measurement largely depends on one well-measured object.

Of the brightest 50 PNe in our sample, only three are contained in M2, and none are among the brightest 16 objects. This suggests that our estimate of the aperture correction may be underestimated. However, as pointed out in Section 5.10, such an error will likely not affect our derived distance to the galaxy. We therefore conservatively assign 0.06 mag as the zero-point error of our PNLF measurements.

The results of our PNLF fit to NGC 4472 are shown in Figure 51. Despite the limited depth of the survey, which restricted the number of PNe in our statistically complete sample to 43, the data are very well fit to the curve of Equation (2). Our distance modulus to NGC 4472 for E(B − V) = 0.02 is therefore ${(m-M)}_{0}={30.85}_{-0.09}^{+0.08}$, or ${14.8}_{-0.6}^{+0.5}$ Mpc.

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For comparison, the almost two dozen SBF distances derived for the galaxy give slightly larger values (30.78 ≤ (m − M)₀ ≤ 31.25) with the most recent value of (m − M)₀ = 31.12 falling on the high side of the interval (Blakeslee et al. 2009). This offset is consistent with the historical difference between the two distance methods. Somewhat curiously, there are no TRGB distance measurements to the galaxy.

Our MUSE-based measurement of ${14.8}_{-0.4}^{+0.3}$ Mpc is slightly more distant, but still consistent, with the first PNLF distance to the galaxy (Jacoby et al. 1990): when scaled to today's zero-point, those authors inferred a value of 14.3 ± 0.4 Mpc. However, our new value is significantly smaller than the results of Hartke et al. (2017), who used PN observations from Subaru's Suprime Cam to derive a distance of 18.1 ± 0.6 Mpc. These on-band/off-band data extend much deeper than the MUSE spectra (to m₅₀₀₇ = 28.8), contain many more (624) PNe, and extend over a much wider field of view than the MUSE observations. We hypothesize that about half the discrepancy is due to a metallicity gradient, with the Hartke et al. (2017) observations principally sampling the bluer outer envelope of the galaxy, while MUSE (and the Jacoby et al. 1990 study) are drawn from the system's inner regions. In a metal-poor environment, a PNLF shift of ∼0.2 mag toward fainter values is possible (see, for example, Dopita et al. 1992).

NGC 5248 is a barred SAB(rs)bc galaxy projected within the Virgo III group, east of the Virgo cluster core. The galaxy's distance is uncertain: while the system has been measured using the Tully–Fisher relation, the results span a wide range of values, from ∼10 Mpc (e.g., Bottinelli et al. 1984; Sorce et al. 2014) to ∼22 Mpc (Ekholm et al. 2000). Given the complexity of the region, none is preferred.

Due to its prominent bar and nuclear ring, the galaxy has been targeted by MUSE as part of the TIMER survey (Gadotti et al. 2019). Neumann et al. (2020) have used these data to create Hα maps of the galaxy's central ring/bar region with the goal of exploring the area's star formation rate history. These authors consider NGC 5248 to be peculiar with respect to other galaxies of its type, with strong star formation in a very large nuclear disk and attached spiral-like features.

We retrieved the archival MUSE-DEEP data cube (ESO Archive ID: ADP.2017-06-14T09:12:09.276, PI: D. Gadotti, Program ID: 097.B-0640) obtained from two observations with an effective exposure time of 3411 s, and 076 seeing at 5007 Å. This pointing is centered on the galaxy's nucleus and includes the system's nuclear ring, two spiral arms, and a number of prominent dust lanes (see Figure 52). Our difference image confirms the presence of strong emission lines throughout the field, and out of an initial sample of 143 candidates, 114 had to be rejected as H ii regions or SNRs (the red circles in the figure). The final sample of PN candidates therefore contains only 29 objects, with 15 being in the top magnitude of the luminosity function.

NGC 5248's cumulative PNLF is plotted in Figure 53. There are two immediate features to note. The first is the luminosity of PN1, which is 0.42 mag brighter than the next brightest object. For galaxies with larger numbers of PNe, such a difference would be highly improbable. Considering NGC 5248's sparsely populated PNLF, though, the possibility that PN1 is normal cannot be discounted.

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The second important property of the distribution is the depth of the survey. If PN1 is overluminous, i.e., if PN1's luminosity is not well described by the luminosity function defined by Equation (2), then the statistically complete sample of PNe only extends ∼0.5 mag down the PNLF. While this lack of depth does not preclude a PNLF distance from being obtained, it does lessen the reliability of the result.

Given the complex morphology of NGC 5248's inner regions, we did not attempt to derive the galaxy's brightness distribution from the literature; instead, we used the information contained within the MUSE data cube to estimate the continuum brightness at the location of each PN. Similarly, since the line-of-sight velocity dispersion in the inner regions of the galaxy is much smaller than MUSE's spectral resolution (e.g., Fabricius et al. 2012; Rosado-Belza et al. 2020), we simply assume σ = 75 km s⁻¹ throughout the region.

As stated above, the only previous distance estimates for this galaxy come from the Tully–Fisher relation. Only one of these determinations places the galaxy beyond 17 Mpc (Ekholm et al. 2000), and most prefer distances on the nearside of Virgo (e.g., Bottinelli et al. 1984; Theureau et al. 2007; Sorce et al. 2014). Our PNLF values are clearly inconsistent with such a location; they agree more with the larger distances.

If we include PN1 in our PNLF analysis, then for an E(B − V) = 0.02, our maximum-likelihood analysis yields a distance modulus of ${(m-M)}_{0}={31.37}_{-0.13}^{+0.06}$, while assigning less than a 0.4% probability that the overluminous PN is a blend of two sources. However, if we accept that the object is not the result of a superposition and adopt the galaxy's most likely distance modulus, then a K-S test excludes the possibility that the observed PNe are drawn from the distribution defined by Equation (2) at the 99% confidence level. Alternatively, if we remove PN1 from the sample, then the distance to NGC 5248 increases to ${(m-M)}_{0}={31.80}_{-0.10}^{+0.07}$ and the hypothesis that the PNe are drawn from the empirical function cannot be refuted. On that basis, we adopt the larger distance to the galaxy. The distribution of likelihoods for this assumption is shown in Figure 54.

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Since the MUSE data cube of the galaxy contains a bright star, the error associated with the photometric aperture correction is very small, formally 0.005 mag. Thus, the systematic component of the error budget is dominated by the ∼0.03 mag uncertainty in the MUSE flux calibration. Our derived distance to the galaxy is then ${(m-M)}_{0}\,=\,{31.80}_{-0.11}^{+0.08}$, or ${22.9}_{-1.1}^{+0.8}$ Mpc.

NGC 6958 is an isolated early-type galaxy, classified as S0 according to Sandage & Bedke (1994) and Laurikainen et al. (2010), but misclassified as a cD by de Vaucouleurs et al. (1991). Thater et al. (2022) used MUSE data of the galaxy to measure the mass of its supermassive black hole via stellar kinematics. Their study also used the dynamical scaling relations for early-type galaxies to place the system at a distance of 35 ± 8 Mpc. If this is correct, then a PNLF measurement would be challenging. On the other hand, it would also offer us an opportunity to apply the technique outside the Local Supercluster in a region where the Hubble flow is relatively unperturbed (see Section 6.4).

We retrieved the data cube of a science verification observation from the ESO Archive (ID: ADP.2017-11-09T16:25:30.895, PI: D. Krajnovic, Program ID: 60.A-9193). The metadata for the observation lists an exposure time of 1906 s and a seeing of 096. However, we have sometimes found discrepancies between the recorded metadata and our own PSF analysis, and indeed, Thater et al. (2022) measured the data cube's PSF FWHM to be 075 at 5000 Å, and ≤06 at wavelengths longward of 7400 Å. Unfortunately, this adaptive optics-supported observation was secured under poor weather conditions, so the data were not optimal.

Despite this shortcoming, we were able to detect 64 PN candidates with our DELF technique. Further inspection confirmed that 28 had spectra consistent with that of a PN. The fact that more than half of the initial candidates appear to be either H ii regions or SNRs (red circles in Figure 55) supports the classification of the galaxy as a lenticular, rather than a cD system. Furthermore, the difference image reveals rotation or outflow of the diffuse emission component.

Figure 56 shows the PNLF of NGC 6958. Clearly, the data are limited: despite the bin size being increased to 0.25 mag, no interval contains more than seven objects. Similarly, as the data only extend ∼0.5 mag down the luminosity function, the shape of the exponential cutoff is ill-defined, which means the PNLF technique may be susceptible to systematic errors. Most importantly, even for the brightest objects, the photometric uncertainties associated with the PN measurements are greater than 0.15 mag. In other words, the S/N of even the brightest detections is only ∼7. Thus, incompleteness may be non-negligible, even at the brightest magnitudes.

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Nevertheless, we fit the data to Equation (2), as per the previous galaxies. Since NGC 6958 has no published surface photometry or kinematic measurements, we used the MUSE continuum spectroscopy for the former and adopted 75 km s⁻¹ for the line-of-sight stellar velocity dispersion. As a lenticular system with an inclination of ∼45°, the latter value is a reasonable estimate for the stellar kinematics. We also excluded all data from the central ∼11'' of the galaxy, where the surface brightness was too bright for PN detections. This left us with V ∼ 12.2 of galaxy light to study.

Figure 57 shows the results of our analysis. Two issues stand out. The first is that, for the best-fit solutions, α_2.5, the PN per unit light ratio, is lower than any other galaxy considered in this study. This suggests that we are missing PNe, even at the bright end of the luminosity function. In such a situation, the PNLF tends to overestimate distance. The second issue is that our distance determination is not very precise, with the 3σ error contours extending over almost two magnitudes in distance modulus. Again, this is due to the small number of PN detections and the limited depth of the survey.

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Formally, our distance to the system is ${(m-M)}_{0}\,=\,{32.80}_{-0.21}^{+0.24}$ (${36.2}_{-3.3}^{+4.3}$ Mpc) for a foreground reddening of E(B − V) = 0.04. Perhaps fortuitously, this value is nearly identical to the Tully–Fisher distance of 36.4 Mpc published by Theureau et al. (2007). Nevertheless, it is fair to conclude that with better image quality and longer exposure times, this object would be well within reach of the PNLF technique. Although our distance exhibits larger errors than usual and may be a slight overestimate due to the underpopulated bright end of the PNLF, it is one of the few targets in this study with a very good aperture correction star. Thus, it is a good candidate for a future program for a PNLF measurement of the Hubble constant (see Section 6.4).

MCG-06-08-024, also listed as FCC 090, is an early-type dwarf galaxy in the Fornax cluster with a Hubble classification of SA0 (Figure 58). It is known from ALMA observations to harbor a disturbed gas reservoir with a molecular gas tail (Zabel et al. 2020). Distance estimates from the globular cluster luminosity function and SBFs range from 17–20.5 Mpc, consistent with the system being a Fornax cluster member. We note that FCC 090 is not listed in the sample of 21 Fornax early-type galaxies whose PNLF was studied by Spriggs et al. (2021); hence, it could be an interesting test case to probe the superior sensitivity of our technique. However, as a dwarf galaxy, we would not expect the galaxy to contain a large population of PNe.

We examined a MUSE-DEEP data cube (ESO Archive ID: ADP.2018-03-26T15:02:26.516, PI: M. Sarzi, Program ID: 296.B-5054) created from two exposures with a total effective exposure time of 4821 s and 067 seeing. From an original list of 13 pointlike [O iii] sources, six were eliminated based on their spectra. The two brightest objects, with m₅₀₀₇ = 27.96 and 28.54, have magnitudes that are roughly 1 mag fainter than the PNLF cutoff of other Fornax cluster galaxies. This precludes us deriving a useful distance estimate but it does allow us to place an upper limit on the system's distance of <30.8 Mpc.

With few exceptions, our PNLF distances agree very well with those of other PNLF results. We already have commented on one major disagreement with the Scheuermann et al. (2022) distance to NGC 1385 and a milder disagreement with the Hartke et al. (2017) measurement of NGC 4472.

A summary of our distances and those based on other methods is given in Table 10. Unfortunately, there is limited overlap between our heterogeneous set of galaxies culled from the MUSE archive and systems with reliable distances found by other techniques. NGC 4038/39 is the only galaxy in our sample with a Cepheid distance, and our PNLF analysis yields a distance that is 10%–27% larger, depending on which Cepheid value is selected (Riess et al. 2011; Fiorentino et al. 2013; Riess et al. 2016, 2022); we note, though, that the most recent Cepheid result yields the least disagreement. That is, the difference between the PNLF and Cepheid distances is similar to that of the Cepheid distances against themselves. Consequently, it is difficult to make any conclusions from the discrepancy.

Table 10. PNLF Galaxy Distances

Galaxy	Number of PNe a	(m − M)0	PN Distance	Other Distance	Notes
			(Mpc)	(Mpc)
NGC 253	34/10	${28.66}_{-0.28}^{+0.12}$	${05.4}_{-0.6}^{+0.3}$	3.5 ± 0.1 (TRGB)	Possible dust within the galaxy
NGC 1052	86/50	${31.26}_{-0.08}^{+0.07}$	${17.9}_{-0.6}^{+0.3}$	18.0 ± 2.4 (SBF)	...
NGC 1326	55/20	${31.00}_{-0.13}^{+0.09}$	${15.9}_{-0.9}^{+0.6}$	...	Overluminous PN rejected
NGC 1351	102/45	${31.39}_{-0.08}^{+0.04}$	${19.0}_{-0.7}^{+0.4}$	19.2 ± 0.6 (SBF)	Overluminous PN rejected
NGC 1366	22/13	${31.39}_{-0.23}^{+0.11}$	${19.0}_{-1.8}^{+1.0}$	21.1 ± 3.0 (SBF)	...
NGC 1385	78/54	${31.99}_{-0.12}^{+0.11}$	${25.0}_{-1.5}^{+1.4}$	...	...
NGC 1399	232/164	${31.23}_{-0.07}^{+0.06}$	${17.6}_{-0.6}^{+0.5}$	21.1 ± 0.7 (SBF)	...
NGC 1404	124/64	${31.37}_{-0.08}^{+0.05}$	${18.8}_{-0.6}^{+0.4}$	18.7 ± 0.6 (TRGB), 20.4 ± 0.6, 20.2 ± 0.6 (SBF)	...
NGC 1419	21/12	${31.39}_{-0.27}^{+0.12}$	${18.9}_{-2.5}^{+1.1}$	22.9 ± 0.9 (SBF)	...
NGC 1433	258/160	${31.42}_{+0.07}^{-0.08}$	${19.2}_{-0.7}^{+0.7}$	...	Two overluminous PNe included
NGC 1512	210/144	${31.43}_{-0.06}^{+0.06}$	${19.3}_{-0.6}^{+0.5}$	11.7 ± 1.1 (TRGB)	Overluminous PN rejected
NGC 2207	3/0	...	<40	...	...
NGC 3501	6/0	...	<38	...
NGC 4038/9	228/154	${31.82}_{-0.07}^{+0.07}$	${23.1}_{-0.8}^{+0.7}$	21.6 ± 0.5 (TRGB)	Antennae galaxies
...	...	...	...	18.1 ± 0.9, 20.4 ± 0.6, 21.0 ± 1.2 (CEPH)	...
NGC 4365	64/30	${31.55}_{-0.09}^{+0.06}$	${20.4}_{-0.8}^{+0.6}$	23.1 ± 0.8 (SBF)
NGC 4418	47/24	${32.59}_{-0.10}^{+0.07}$	${32.9}_{-1.5}^{+1.2}$	...	Overluminous PN included
NGC 4472	67/43	${30.85}_{-0.09}^{+0.08}$	${14.8}_{-0.6}^{+0.5}$	15.9 ± 1.0 (SBF)	SBF is an average of 22 values in NED
NGC 5248	29/15	${31.80}_{-0.11}^{+0.08}$	${22.9}_{-1.1}^{+0.8}$	...	Overluminous PN rejected
NGC 6958	28/15	${32.80}_{-0.21}^{+0.24}$	${36.2}_{-3.3}^{+4.3}$	...	...
MCG-06-08-024	7/0	...	<30.8	...	...

Note.

^a Given as the total number of PNe/approximate number of PNe contributing to distance.

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A comparison of the distance to the six galaxies with TRGB measurements is equally problematic. The PNLF distance to the edge-on spiral NGC 253 is ∼54% greater than that found from the system's red giants; however, the sparseness of the luminosity function (less than 10 objects within 1 mag of M*) and the possible effects of internal extinction, weaken confidence in the PNLF result. Conversely, for the more face-on spirals of NGC 1433 and 1512, the factor of ∼2 discrepancies between the PNLF and TRGB distances (Sabbi et al. 2018) is likely due to the galaxies' large population of AGB stars and the misidentification of the RGB tip. As for the three remaining galaxies, NGC 1404 (Hoyt et al. 2021), NGC 4038/9 (Jang & Lee 2015), and Paper I's measurement of NGC 628 (Sabbi et al. 2018) the two distance methods agree within +0.5%, +7%, and +8%, respectively, where the PNLF distances are always larger. This more encouraging result is consistent with previous TRGB–PNLF comparisons (e.g., Ciardullo 2012, 2022).

The richest area for a PNLF comparison is with the SBF method where we have 10 galaxies in common, eight in this paper and two in Paper I. On average, the PNLF distances are smaller by −6%, a systematic that has been known for decades and has usually been attributed to the effect of internal extinction in the calibrating galaxies (Ciardullo et al. 1993, 2002). Briefly stated, the PNLF and SBF respond to undetected attenuation in opposite ways: distances from the PNLF are underestimated, while those from the SBF are overestimated (due to the color term associated with the fluctuation magnitude). Since both methods are calibrated in the bulges of nearby spirals, but then applied to elliptical and lenticular systems, any internal attenuation in the calibration galaxies that is not present in the program systems will lead to a systematic offset in the distance scales. Notably, the −6% offset is less than that found in the older studies, lowering the requisite amount in attenuation in spiral bulges to less than E(B − V) = 0.01 mag.

Scheuermann et al. (2022) presented another approach to comparing PNLF distances to other methods in their Figure 9. In comparison to Cepheids and TRGB, their PNLF distances tended to be too large by ∼7%, but with significant scatter, due in large part to limited repeatability within each of those methods.

While it is premature to declare that the PNLF compares as well to other premium distance techniques as they do to each other, the results are quite promising. For the PNLF to be validated further, it is essential to obtain MUSE data cubes taken with absolute photometry of faint point sources in mind. This means (a) choosing fields with foreground stars bright enough to define the observation's PSF and aperture correction, (b) observing when the atmospheric seeing is good enough to support faint PN identifications, (c) ensuring that each data cube is taken under the best photometric conditions, and (d) including enough galaxy luminosity such that, with the proper exposure time, at least 30 PNe will be detected in the top 1 mag of the luminosity function.

In addition, the PNLF technique can benefit from additional methodological improvements that we summarize in 7.

In Paper I, we described several classes of objects that could be confused with PNe and the ways in which MUSE spectra can be used to reject them from our PN samples. The possible interlopers that were discussed included SNRs, H ii regions, and background galaxies. Not mentioned in the discussion was another possible source of overluminous PNe that can cause confusion. As first noted by Jacoby et al. (1996), Wolf-Rayet (W-R) nebulae (Nazé et al. 2003) can possibly contaminate the PNLF in star-forming galaxies. Because some of these high-mass nebulae are excited by very luminous O-stars with temperatures of ∼100,000 K (similar to the central stars of PNe), their spectra can be indistinguishable from PNe (Chu 2016). Yet the [O iii] luminosities of these PN mimics can be nearly 10 times brighter than M*. Figures 1(b) and 4(b) of Esteban et al. (1994) show spectra of two W-R nebulae in M33. At large distances, these objects can appear pointlike and can contaminate the PNLF.

Fortunately, in many cases, this source of confusion should not be a problem. Specifically,

• 1.  
  In classic E/S0 galaxies, Pop I objects such as W-R nebulae are unlikely to exist.
• 2.  
  In star-forming galaxies closer than ∼20 Mpc, the continuum from the nebula's exciting O-star (M_V as luminous as −7) can be bright enough to detect (Nazé et al. 2003; Kehrig et al. 2011). Additionally, in good seeing, the nebulae will generally be large enough (r > 20 pc) to be spatially resolved.
• 3.  
  In systems where W-R nebulae are relatively rare, the objects will likely either be much more luminous than the PNLF cutoff, and be identifiable as outliers, or fainter than M* and have a negligible effect on the PNLF distance. It is only when W-R nebulae are marginally brighter than M* in star-forming galaxies beyond Virgo and Fornax that they can influence the PNLF maximum-likelihood fit. In such galaxies, the line diagnostics of W-R nebulae will place them on the border between H ii regions and PNe, and thus make them hard to distinguish from true PNe. Among the galaxies in this study, PN1 in NGC 5248 is a possible candidate for a W-R contaminant.

There may also be unusual emission-line sources, such as the ultraluminous X-ray source in Holmberg-II (Lehmann et al. 2005) and the black hole in the NGC 4472 globular cluster RZ 2109 (Zepf et al. 2008) that could be mistaken for PNe. At the distance of NGC 1385 (d ∼ 25 Mpc), these objects, which are as bright or brighter than M*, would have m₅₀₀₇ ∼ 24–25. Fortunately, such objects are rare and can have spectral features that discriminate them from PN. Thus, they are unlikely to be a problem.

PNLF distances rely on fitting the apparent magnitude distribution of PNe within a galaxy to some assumed form for the distribution of absolute PN magnitudes. This paper uses the expression first proposed by Ciardullo et al. (1989) as a template, but other forms of the function are possible (e.g., Longobardi et al. 2013; Valenzuela et al. 2019). The better we can define the true shape of the PNLF, the better the results of the fit.

To that end, it is instructive to sum the absolute [O iii] magnitudes of all the galaxies in this survey to produce a single grand PNLF. In theory, such an analysis could lead to an improved expression for the shape of the PNLF cutoff: by reducing the errors introduced by counting statistics, subtle deviations from Equation (2) could become apparent. On the other hand, any error associated with the conversion of apparent to absolute magnitude will propagate into the summed PNLF; such errors include the formal uncertainties of the galaxy distance determinations, the field-to-field zero-point errors for systems with more than one MUSE data cube, and any population-dependent change in the PNLF's shape. These errors will all work to soften the abruptness of the function's bright-end cutoff, (slightly) flatten the PNLF's power-law slope, and smooth over possible high-frequency features in the distribution.

Table 11 lists the galaxies analyzed in this paper that have well-defined PNLFs. Also listed in the table are NGC 628 (Paper I) and NGC 1380 (Paper I and Chase et al. 2023), both of which also have high-quality DELF measurements. Included in the table are our estimates for the galaxies' absolute λ5007 magnitudes for 90% completeness (assuming their most likely PNLF distances), the total number of PNe detected, and the number of PNe above the completeness limit. These data are coadded into the single PNLF shown in Figure 59.

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Table 11. Combined PNLF

Galaxy	${M}_{\mathrm{lim}}$	Total PNe	PN with $M\lt {M}_{\mathrm{lim}}$
NGC 628	−3.19	202	90
NGC 1052	−3.74	87	39
NGC 1326	−3.26	55	39
NGC 1351	−3.43	102	45
NGC 1380	−3.22	112	78
NGC 1385	−3.65	78	40
NGC 1399	−3.27	232	163
NGC 1404	−3.70	126	38
NGC 1433 a	−3.25	258	95
NGC 1512	−3.16	210	144
NGC 4038/9	−3.40	228	152
NGC 4365	−3.71	64	30
NGC 4418	−3.86	47	24
NGC 4472	−3.91	81	39

Note.

^a PNe from Pointings P2, P3, and P5 have been excluded, due to their poorly determined aperture corrections.

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Before discussing the luminosity function, it is imperative to be aware of the nuances involved in performing this coaddition. The solid curve attempts to model the effect of distance uncertainties in the data set by smoothing our adopted PNLF (Equation 2) by kernels derived from the galaxies' likelihood contours. While the resultant model is a better fit to the coadded PNLF, there is still a slight overdensity of objects brighter than M*, and there are a number of other factors that are not included in the analysis. For example, zero-point errors caused by uncertain MUSE flux calibrations, aperture corrections, foreground dust, and possible metallicity dependencies will not affect the coaddition for galaxies observed in a single MUSE pointing. However, these zero-point issues will distort the PNLF's shape in systems observed via a mosaic of MUSE frames (see Section 5.10). The propagation of that error into the summed PNLF is not being modeled.

Similarly, a simple coaddition does not account for the presence of PN superpositions, i.e., two (or more) pointlike [O iii] sources projected onto the same resolution element of the detector. When blends are present, they can distort the shape of the PNLF and produce a distance solution that is bimodal; the likelihood contours shown in Figure 36 provide one such example.

In this study, we have minimized the effect of blends by excluding PNe within a few arcsecs of the parent galaxy's nucleus. As a result, even when possible super-M* objects were detected, the likelihood that they were caused by blends was relatively low (see Figures 10, 36, and 48). However, while the probability of such an occurrence in any one galaxy may be small, the likelihood of a superposition being present somewhere in the data set is significant. The curve shown in Figure 59 does not account for this possibility.

Another effect not modeled is the effect of dust within the target galaxies. Internal reddening in late-type galaxies may distort the shape of PNLF by effectively subdividing the PN population into multiple components: some extincted and some not. Models suggest that the effect of this bifurcation (or trifurcation) on the PNLF distances is minor (e.g., Feldmeier et al. 1997; González-Santamaría et al. 2021; Guo et al. 2021), but in highly inclined or interacting systems, it may have an effect.

Finally, it is important to note that our coaddition is only sensitive to changes in the PNLF's shape among different galaxy populations. Any process that shifts the location of the PNLF cutoff, but not the function's shape, will not be detected by the analysis. In other words, Figure 59 only shows that the shape of the bright end of the PNLF depends very little on the type of galaxy being observed; it does not provide any information about whether, for example, the PNLF of spiral galaxies are systematically brighter than those of early-type systems.

The PNLF of Figure 59 consists of 1016 objects brighter than their galaxies' completeness limits. Overall the data follow the prediction based on Equation (2) curve extremely well. The two noticeable deviations are a possible slight overestimate of the number of PNe with M₅₀₀₇ = −3.85 and a clear underestimate of objects with magnitudes brighter than M*.

The latter feature is critical for fitting the PNLF, but the significance of the feature is difficult to determine. As pointed out above, the curve showing the expected PNLF does not include a number of effects such as PN superpositions and zero-point differences in a mosaic of MUSE pointings. These issues could easily produce the high-luminosity tail seen in Figure 59.

Another possible cause of the PNLF's high-luminosity tail is contamination by non-PNe. Line diagnostics are very efficient at eliminating objects such as H ii regions and SNe remnants from the PN sample. Still, some interlopers may slip through, including nebulae produced by W-R stars (see Section 6.2). A careful investigation of the super-M* objects shown in Figure 59 is beyond the scope of this paper but clearly needs to be done.

Of course, the high-luminosity tail of Figure 59 may be real. Davis et al. (2018) showed that the true [O iii] luminosities of PNe in M31's bulge extend well beyond M*. They argue that the observed bright-end cutoff of the PNLF is a product of the dust produced during the AGB phase: the more massive the core, the more massive and dusty the envelope, and the greater the circumnebular extinction. This hypothesis is consistent with the results of Ciardullo & Jacoby (1999), who found a steep correlation between the core mass of a PN and its self-extinction. If this is true, then the viewing angle may affect the observed brightness of a distant PN; for instance, a favorable orientation (e.g., along a PN's polar axis) may result in less attenuation and a brighter apparent magnitude. Consequently, an extremely luminous PN, when viewed at just the right angle, may be recorded as overluminous. In this cartoon model, the bright-end tail seen in Figure 59 is intrinsic to the PNLF itself, and Equation (2) must be modified to include the existence of these objects.

The PNLF method has been improved significantly with VLT/MUSE and the DELF techniques. With a sufficient sample of well-observed and well-selected galaxies, the method can help deliver a measurement of H₀, either directly via distances to galaxies in the relatively unperturbed Hubble flow, or by calibrating SN Ia luminosities in the spirit of the Cepheid and TRGB approaches.

The zero-point of the PNLF distance scale, though, cannot be tied to stars in the Milky Way. Thus, their distances are not completely independent of the results of other methods (e.g., Cepheids, eclipsing binaries). However, the PNLF works in all types of large galaxies, from the latest Pop I systems to the reddest elliptical galaxies, and the technique can reach systems outside the Local Supercluster that are ill-suited for Cepheid, TRGB, and/or SN Ia studies. Moreover, the requisite data can be acquired quickly and efficiently from the ground, without the need for expensive space-based observations. Thus, a PNLF measure of H₀ can offer a unique perspective on the robustness of other results.

A PNLF program to measure H₀ would have to target galaxies that:

• 1.  
  are relatively isolated, in order to minimize the effect of peculiar motions on the estimation of the Hubble flow. The larger the galaxy group, the larger the number of PNLF distances that would be needed to obtain a true measure of the system's Hubble velocity and distance.
• 2.  
  have distances exceeding ∼30 Mpc to reduce the effects of flows within the local volume of space.
• 3.  
  have absolute V-band magnitudes of M_V ≲ −19.5 (i.e., V absolute luminosities of at least ∼5 × 10⁹ L_⊙). Any PNLF study must survey at least this amount of light to ensure that the top ∼1 mag of the PNLF is sufficiently populated to deliver a precision measure of the PNLF cutoff. In practice, a target galaxy must be brighter than this, since (a) observations from a limited-field instrument such as MUSE will likely not cover the entire galaxy, and (b) PNe projected in the highest surface brightness regions of the system will be difficult to detect.
• 4.  
  have moderately bright point sources in the field to enable a precise measurement of the photometric aperture correction. Foreground Milky Way stars are ideal, but for systems beyond ∼30 Mpc, bright globular clusters will suffice.

In addition, although the PNLF can be applied to galaxies of all Hubble types, the early-type systems are most easily analyzed, as their PN samples will not be severely contaminated by interlopers such as H ii, SNRs, and W-R nebulae. Uncertainties associated with internal extinction would also not be an issue.

With those caveats in mind, of the 23 galaxies analyzed in this paper and Paper I, only six are potentially useful for an H₀ study, and only two have distances greater than 30 Mpc: NGC 6958, which is poorly measured by the present observations, and NGC 4418. At face value, the H₀ values found from these two objects are 69.2 ± 11.0 and 75.0 ± 9.6 km s⁻¹, respectively; their weighted average is 74.2 ± 7.2 km s⁻¹ Mpc⁻¹. Note that because the quoted errors are dominated by the uncertain (∼10%) contribution of peculiar motions to the galaxies' recessional velocities, the error on H₀ can be reduced to <4% with a larger sample (∼25) of galaxies.

It should be stressed that most of the galaxies mentioned in this paper were not observed in a manner optimal for distance scale studies. NGC 4418 happened to have a bright foreground star present in its MUSE data cube, but most of the systems discussed in this paper do not. It is also possible that some of the galaxies were observed during nights of uneven transparency. And most notably, the effect of metallicity shifts on the PNLF has not been considered. For H₀ studies, we expect this last effect to be negligible: although the PNLF cutoff is known to fade slightly in populations with sub-LMC metal abundances (e.g., Ciardullo & Jacoby 1992; Dopita et al. 1992; Schönberner et al. 2010), such systems are not generally the target of a PNLF investigation. As pointed out above, the PNLF cutoff is not well defined in systems fainter than M_V ∼ −19.5. According to the mass–metallicity relation for local galaxies (e.g., Tremonti et al. 2004), this basically ensures that any system measured well enough to be useful for a study of the Hubble constant will be in the regime where metallicity effects are minimal.

The fact that our H₀ result is very similar to current best estimates in the Hubble tension literature is encouraging, but likely fortuitous. Clearly, in order to advance the use of the PNLF for distance scale studies, we need a larger sample of galaxies that meet the criteria listed above, and this means new targeted observations are necessary.

Distances derived using the PNLF method have generally been highly consistent across authors, reflecting only the stated errors in the measurements. Thus, the discrepancy between the distance to NGC 1385 derived in this paper and that found by Scheuermann et al. (2022) is very unusual and warrants exploration. We first consider whether NGC 1385's properties are consistent with membership in the Eridanus cluster 20–25 Mpc away. We then look in detail at the PN photometric results to uncover any clues.

The plausibility of NGC 1385's distance determined in this work, compared to the much lower Tully–Fisher values from the literature, and the distance derived by Scheuermann et al. (2022), can be assessed through its probable membership in the Eridanus cluster. Table 12 compares the two PNLF distances to SBF measurements of Eridanus cluster members. Our distance of 25.0 Mpc is consistent with cluster membership, and the galaxy's radial velocity of 1499 km s⁻¹ is close to the mean of the cluster (see Figure 5 of Willmer et al. 1989). Based on these data, the Scheuermann et al. (2022) distance, which is less than half our value, would seem implausible.

A composite of the [O iii] images for the five MUSE pointings, labeled P1–P5, is shown in Figure 60. Our detections are indicated with blue markers, and the PNe are numbered individually for each pointing, starting with PN1. Point sources that were rejected as either H ii regions or SNRs are marked with red circles without labels. PNe from the sample of Scheuermann et al. (2022) are plotted in magenta and their SNRs are shown in orange. For the following detailed comparison, it is useful to remember that our brightest PN has an [O iii] magnitude of m₅₀₀₇ = 27.55.

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Pointing P1. Our work yielded 24 pointlike [O iii] sources with spectra consistent with that of a PN; these sources have a brightness range between 27.55 ≤ m₅₀₀₇ ≤ 29.54 mag. In contrast, Scheuermann et al. (2022) find only three objects in the field that they classify as PNe; their [O iii] magnitudes are m₅₀₀₇ = 26.52, 26.65, and 27.25. The three identifications are detailed below.

PN6. Our analysis suggests that the data reduction processes used by the ESO Archive are slightly different from those employed by Scheuermann et al. (2022), and this results in a small astrometric offset between the two data sets. Thus, we strongly suspect that the object that Scheuermann et al. (2022) refer to PN6 as an unlabeled source in Figure 60, which has a red marker within the magenta marker (arrow). This places the Scheuermann et al. (2022) source 080 east and 093 north of our position. We classify the object as an unresolved, high-excitation H ii region, as its [S ii] line ratio diagnostic is at the low-density limit.

PN7. The object is located in a giant H ii complex, and we classify its spectrum as that of an H ii region.

PN11. Like PN7, the object is located in a giant H ii complex, and based on its spectrum, we classified it as an H ii region.

In summary, we classify none of Scheuermann et al. (2022) objects as PNe. Conversely, none of our 24 DELF-detected PNe are reported by Scheuermann et al. (2022).

Pointing P2. We find 23 [O iii] sources with spatial and spectral properties consistent with those of PNe. The brightnesses of these objects range between 27.58 ≤ m₅₀₀₇ ≤ 29.14. In contrast, Scheuermann et al. (2022) report five PNe with magnitudes of m₅₀₀₇ = 25.72, 26.47, 27.63, 27.90, and 27.97 mag. These Scheuermann et al. (2022) identifications are as follows.

PN1. This object, which is located within a giant H ii complex, is extremely bright in [O iii], but its [S ii] lines indicate a low-density nebula. If the object were a PN, it would be 1.83 mag (5.4 times) brighter than our next brightest planetary. Moreover, if the galaxy were at the distance of the Eridanus cluster, the luminosity of the PN would imply a central star luminosity of $\mathrm{log}L/{L}_{\odot }=4.48$, which is well above any reasonable post-AGB evolutionary track. We conclude that this object cannot be a PN.

PN5. This source, which is projected on a complex background, is brighter in Hα than it is in [O iii] and thus violates the Herrmann et al. (2008) criterion for [O iii]-bright PNe. As indicated by the [S ii] doublet, the object does have a high nebular density. Nevertheless, it is not classified as a PN.

PN16. The [S ii] lines of this source are half the strength of Hα. It is an SNR.

PN17/PN19. The coordinates of these two sources are almost identical. As we find no evidence for a superposition of two point sources, neither in terms of separation in wavelength nor in terms of an elongated image, we conclude that it is a single object. We agree with Scheuermann et al. (2022) that it is a PN (our PN2).

Pointing P3. We find 16 PN candidates in the brightness range between 27.76 ≤ m₅₀₀₇ ≤ 29.19. Scheuermann et al. (2022) report no detections in this field.

Pointing P4. We find four PN candidates in the brightness range between 27.44 ≤ m₅₀₀₇ ≤ 28.32 mag. Scheuermann et al. (2022) report no detections in this field.

Pointing P5. We find 19 PN candidates in the brightness range between 27.56 ≤ m₅₀₀₇ ≤ 29.09. Scheuermann et al. (2022) find three PNe candidates with brightnesses of m₅₀₀₇ = 27.07, 27.37, and 27.97. Specifically, the Scheuermann et al. (2022) identifications are as follows:

PN10. The spectrum of this object clearly supports its classification as an H ii region.

PN13. Like PN10, the spectrum of this object is consistent with that of an H ii region.

PN18. We agree that this object is a PN. However, our photometry yields a magnitude that is ∼1 mag fainter than that quoted by Scheuermann et al. (2022), i.e., m₅₀₀₇ = 28.84 ± 0.28.

The brightest PNe in a galaxy are the most important objects for deriving a PNLF distance. After a careful review of the brightest PNe reported by Scheuermann et al. (2022), we find that our photometry and spectroscopy do not support those objects' classifications as PNe. Consequently, we are unable to reproduce the short (9.8 Mpc) distance to the galaxy.