Toward Precision Cosmology with Improved PNLF Distances Using VLT-MUSEI. Methodology and Tests

We validated the performance of the DELF technique using data cubes centered on the nuclei of galaxies. These regions have a wide range of continuum surface brightnesses and are therefore excellent locations for testing the efficiency of our photometric techniques. NGC 1380 is a good place to begin as we can compare our own data with S/N values for PNe published by Sp2020. Figure 10 presents the results of this analysis.

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On the left, the graph shows the residual continuum noise in the wavelength interval 5100 ≤ λ ≤ 5500 Å, measured from PN spectra that were extracted with DAOPHOT using an aperture radius of 3 pixels and a sky annulus between 12 and 15 pixels. The noise is plotted in units of 10⁻²⁰ erg cm⁻² s⁻¹ Å⁻¹ for 104 PN candidates of all brightnesses measured on both the unfiltered data cube and the DELF-processed data cube. The objects are located in regions with a variety of surface brightness. As expected, the residual continuum noise is clearly correlated with the background surface brightness. In regions of low surface brightness, the unfiltered and filtered noise levels are the same (the gray 1:1 line). However, the unfiltered noise increases rapidly as the background brightens (gray curve, representing a fourth-order polynomial). The noise levels for the DELF measurements are never worse than those for the unfiltered data, and at high surface brightness, they are far superior. We conclude that the MUSE-specific on-band/off-band technique is the preferred choice for further processing.

This is confirmed by comparing the S/N of PN magnitudes with and without filtering. We have chosen to use the quantity A/rN introduced by Sp2020, which is the ratio of the amplitude of their simultaneous fit of [O iii] in the spatial and spectral domains over the residual noise of the fit. We compute the same ratio in our background-subtracted spectra by fitting a Gaussian to the [O iii] λ5007 line and measuring the noise, again in the wavelength interval 5100 ≤ λ ≤ 5500 Å. The plot on the right in Figure 10 shows A/rN for the unfiltered data versus A/rN with the filter. The color coding is the same as for the previous test. Sp2020 considered objects measured with an A/rN below 3 to be uncertain. Our plot reveals that a considerable fraction of measurements without the filter fall below this threshold. The same measurements with the DELF technique lie well above the Sp2020 threshold. The correlation with background surface brightness is again evident.

Additional plots which directly compare our A/rN and magnitude measurements with those of Sp2020 are presented in Section 4.1. Based on these three analyses, we conclude that DELF is indeed an efficient tool for suppressing systematic flat-fielding errors in MUSE emission-line spectrophotometry. While this is especially important in the inner regions of galaxies where PNe are plentiful and the continuum surface brightness is high, DELF offers significant improvement even when the background surface brightness is low, as the photometric uncertainties will still be dominated by fixed-pattern flat-fielding errors.

In order to check the validity of our photometry, we performed tests with artificial emission-line point sources embedded in the original MUSE data cubes with a priori known positions, fluxes, and radial velocities. Running the algorithms on the original data with these mock PNe allows us to assess random and systematic errors for our photometry and spectroscopy, and determine how the detection limit for a given cube depends on the continuum surface brightness across the galaxy.

Figure 11 illustrates two examples from an extensive series of tests: frame (a) shows the distribution of a regular grid of PN positions, plotted over the halo surface brightness of NGC 1380 in the continuum. Frame (b) presents the image corresponding to the Doppler-shifted wavelength of [O iii], extracted from the simulated data cube over three layers for a better S/N. The grid comprises 11 groups of 11 mock PNe with magnitudes between m₅₀₀₇ = 28.0 and m₅₀₀₇ = 29.0 in increments of 0.1 mag, placed diagonally with decreasing brightness from upper left to lower right. Frame (c) is analogous to (a), but the PNe have random positions, random radial velocities (hence they appear in different data cube layers), and random m₅₀₀₇ magnitudes in the range 27.0 ≤ m₅₀₀₇ ≤ 29.5.

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The mock PNe were inscribed into the data cube as follows. First, a two-dimensional point-source image was created using the assumption of a Gaussian PSF. A Gaussian was chosen because the details of the PSF were deemed to be unimportant, as all our photometry is performed using small (3 pixel) apertures and a sky annulus with inner and outer radii of 12 and 15 pixels. The FWHM and total flux contained in the point source were varied from run to run, with values appropriate for the specific galaxy being analyzed.

Next, the noise-free point sources, which were in units of photons per pixel, were modified with a Poissonian noise generator, using the IDL function POIDEV. These noisy images were then transformed into cgs units and measured with DAOPHOT's aperture photometry routine phot to determine their "true" magnitudes.

Finally, the noisy 2D image was added into the data cube of a galaxy, either in a grid at a common wavelength or at random positions with wavelengths distributed over the 15 data cube layers that are relevant for the galaxy in question (see Section 3.2) and weighted by the Gaussian profile of the assumed MUSE line-spread function (LSF). In other words, the input image with artificial PNe is projected into the stack of 15 data cube layers such that each PN has the correct LSF for its assigned Doppler shift.

It is worthwhile mentioning that sometimes the random assignment of a PN position leads to a chance superposition of objects, e.g., PN82 and PN87 in Figure 11(c), that can be hard to distinguish in a 2D image (d). In some cases, the PNe can be resolved in the data cube by their velocity difference. However, if Δv ≲ 100 km s⁻¹, the PNe may appear as one object. While this seems to be an academic exercise of the simulations, we find such examples exist in reality and have a potential impact on the PNLF (see the discussion in Section 4.1).

The analysis of photometric simulations as dense as the one in Figure 11(c) also reveals that aperture photometry is negatively influenced by the presence of too many emission-line objects in the sky annuli. Such a condition causes complex cross-talk and systematic errors that are hard to remove. Because the density of bright PNe in distant galaxies is generally not high enough to trigger these problems, we conducted the remainder of our simulations using a modified position generator that reduces the number of objects per frame by imposing a minimum distance between sources (30 pixels). This constraint entirely erased the cross-talk artifacts observed in the simulations. It was subsequently chosen as the standard routine.

Using data cubes with artificial PNe placed on a grid as shown in Figures 11(a) and (b), we compared two different aperture photometry tools for internal consistency: DAOPHOT's aper, and IRAF's phot. The results are shown in Figure 12. The overall agreement in a magnitude range of 28.5 ≤ m₅₀₀₇ ≤ 29.5 is very good. We note that this test is idealized in the sense that the centroids of the point sources are accurately known a priori; this would not be the case for measurements of unknown objects and, in particular, very faint objects at the frame limit. However, the exercise is useful to demonstrate the validity of our photometry down to faint magnitudes.

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The lower-right panel in Figure 12 plots the photometric residuals for the two tools, showing average values in 0.2 mag bins. The DAOPHOT results exhibit the best agreement with the input values of the simulation. The error bars on the red symbols illustrate the typical uncertainty of a single measurement in each bin, not the error of the mean, which is plotted with dashed lines. The gray lines represent the envelope for the standard deviation of the DAOPHOT measurements in each bin; these are in reasonable agreement with the error bars of single measurements as taken from the DAOPHOT error estimates. The kink at m₅₀₀₇ = 28.6 is an artifact of our simulation and is due to the systematics of the data cube and the fixed pattern of the grid.

In order to remove the systematic, a fully randomized simulation as highlighted in Figures 11(c) and (d) was executed. Using an automated script, we generated 10,000 artificial PNe and distributed the objects among 200 data cubes, with 50 objects per cube. The results of these models are plotted in Figure 13. The left panel shows a scatter plot of all 10,000 measurements, while the plot on the right illustrates the typical 1σ uncertainties reported for individual measurements. The envelope curves indicate the measured standard deviation in each bin.

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Except for the faintest magnitudes, the quantities agree well, confirming that the DAOPHOT error estimates are statistically meaningful and credible. At a magnitude of m₅₀₀₇ = 27.0, which is slightly fainter than the PNLF's bright-end cutoff for NGC 1380, the simulated photometry indicates individual errors of 0.04 mag. Errors of the mean, which are of order 0.01 mag, demonstrate that there is no systematic error in the photometry down to m₅₀₀₇ ∼ 28.0. Beyond that point, a systematic offset does become apparent, and the amplitude of the systematic grows to ∼0.07 mag at m₅₀₀₇ ∼ 29, which is roughly the detection limit in the cube.

The simulations confirm that our technique yields precision spectrophotometry for point-like emission-line sources having the magnitudes of PNe at distances of 15 ≲ D ≲ 25 Mpc. In typical 1 hr exposures, measurements of the bright end of the PNLF should be accurate to ∼0.04 mag with no apparent systematic errors, even in regions of high surface brightness. The errors can clearly be reduced and the method extended to greater distances with larger PN samples and with longer exposure times. Based on these promising results, we perform all of our photometric measurements with DAOPHOT.

The line-fitting tool of our spectroscopy provides a measurement of the line-of-sight velocities of individual PNe and thus allows for future exploration of the gravitational potentials of PN host galaxies. To test this capability, we used our simulations to assess the accuracy of the Gaussian fit to the [O iii] emission line. Figure 14 shows the velocity residuals of mock data, obtained from 1000 realizations of PNe in a total of 20 data cubes, each simulating a 1 hr MUSE exposure of a galaxy. The full range of velocity residuals is the equivalent to two MUSE wavelength bins, i.e., 2.5 Å. It is immediately apparent that the velocity accuracy is on the order of 1/10 of a wavelength bin, with a standard deviation of 5.0 km s⁻¹ for objects in the magnitude range 27.0 ≤ m₅₀₀₇ ≤ 27.5, 7.0 km s⁻¹ between 27.5 ≤ m₅₀₀₇ ≤ 28.0, 11.1 km s⁻¹ between 28.0 ≤ m₅₀₀₇ ≤ 28.5, and 16.4 km s⁻¹ for PNe fainter than 28.5 mag. The error of the mean again demonstrates that there is no systematic offset in the measurements. The simulation shows that the central wavelength error is well behaved, and that radial velocities at the bright end of the PNLF can be measured with an accuracy of a few km s⁻¹. We have used this result in our subsequent analysis of benchmark galaxies for calibrating the line-of-sight velocity (LOSV) error as a function of PN brightness.

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We retrieved all MUSE exposures in the vicinity of NGC 1380 from the ESO archive. This consisted of 43 frames taken during the time-frame between 2016 December 30 and 2017 November 10 from program ID 296.B-5054 (PI: M. Sarzi). As displayed in Figure 15, these data consist of pointings in three fields with a small amount of spatial overlaps between fields. We used the master calibrations from the archive and chose to only re-reduce the science data (using pipeline v2.8.3, Weilbacher et al. 2020) as follows. We processed the basic calibrations using bias correction, flat-fielding, tracing, wavelength calibration, geometric calibration, and twilight sky-flat correction using the master calibration closest in time to each science exposure.

The high-level processing then handles the data at the level of individual exposures. We first reduced the offset sky fields to measure the sky continuum and produce a first guess for the sky-line fluxes. For each on-target exposure, the pipeline then combined the data from all CCDs while also correcting for atmospheric refraction. The flux calibration used response curves (and telluric corrections) derived from standard star exposures taken during the same night, typically, within an hour of the science exposure. All response curves were visually checked to be valid in the wavelength range below 7000 Å; the curves showed only typical night-to-night variations. The pipeline then refit the sky emission lines, subtracted them together with the previously prepared sky continuum, corrected the data to the appropriate barycentric velocity, applied the relative astrometric calibration, and finally created a data cube and set of broadband images of each exposure (including an image integrated over the bandpass of HST F814W filter). Automatic alignment of the exposures failed, because the fields of NGC 1380 contained significant background gradients and the foreground stars were relatively faint. We therefore subtracted the large-scale gradients using smoothed images and then interactively computed the stellar centroids in each MUSE image and on the HST F814W reference image using the IRAF routine imexam. While doing so, we used the frame FWHM given by IRAF to remove exposures with bad seeing. The exposures selected for the final cubes are listed in Table 1.

Using the above astrometric offsets, we then used the pipeline again to combine the good exposures of each field into a data cube. The cube reconstruction rejects cosmic rays, and we saved the data in the default sampling (02 × 02 × 1.25 Å) with the wavelength scale starting at 4600 Å. A comparison of the positions of six stars from the Gaia DR2 catalog (Lindegren et al. 2018) with those derived from our MUSE image in the F814W filter shows nonnegligible but approximately random offsets at about the 007 level. We therefore conclude that the MUSE cubes have an astrometric accuracy on the same order. ⁷

As the first step of the analysis, each cube produced by the MUSE data reduction was processed with the DELF filter to yield two diff files, one containing 13 layers of 3 coadded wavelength bins (used for source detection), and the other containing 15 layers of unbinned data (for PN measurements).

The CENTER, MIDDLE, and HALO fields were inspected visually with DS9 as described in Section 3.2. This step, which yielded 162, 73, and 29 PN candidates, respectively, is illustrated in Figure 16. In addition, Figure 3 in Section 3.2 shows our CENTER pointing in the continuum and in two narrow layers of the stack of 13 coadded images. This figure highlights how emission-line objects appear and disappear on opposite sides of the nucleus, owing to the rotation of the galaxy, and the associated Doppler shift of the PNe. The northern part of NGC 1380 is rotating toward us (blueshifted), while the southern part is moving away from us (redshifted). Also, in addition to the point sources, there is also a prominent feature seen near the nucleus of the galaxy which suggests the presence of an ionized disk. This disk is likely associated with the dust ring seen in the inset of (a) and participates in the rotation of the stars. We discuss this object briefly in Section 4.1.4.

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The careful double-checking of diff images, e.g., Figure 3, occasionally reveals a small mismatch in the continuum scaling factor. In this example, the mismatch is apparent to the north and south of the nucleus as white hues. This less than perfect subtraction, which is caused by rotation-induced Doppler shifts of the stellar population, is irrelevant for the point-source photometry, which uses local estimates of the background.

We display the order of magnitude mismatch of the diff image in Figure 17. The figure shows noise histograms for ten 51 × 51 pixel regions of the bluest unbinned diff layer in the CENTER pointing of NGC 1380. The histograms start at pixel (70, 160) and then moving outward in the galaxy in increments of 10 pixels in x. The data show an almost perfect normal distribution of noise that increases toward the nucleus as the surface brightness of the galaxy rises. The mean varies by an amount of less than 3 × 10⁻²⁰ erg cm⁻² s⁻¹, which is negligible in comparison with the flux of PNe near the detection limit (≈300 × 10⁻²⁰ erg cm⁻² s⁻¹).

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It is worth pointing out that the detection of PNe turned out to be an iterative process, involving several passes through imaging, photometry, and spectroscopy. A detailed inspection of the images was required, which led to the discovery of as many as 15 point sources with overlapping images but different line-of-sight velocities. These blended objects were then confirmed by carefully stepping through the stack of images. A full record of detected blends is listed in Appendix B, Table 5.

Figure 18 illustrates an example of such a blend. In the figure, three PN candidates are located within a region less than 1'' in radius; such a blend would have been impossible to distinguish using the classical narrowband filter technique. Of these three sources, two were detected by Sp2020: one was classified as a PN (object Sp68 in their Table 4) and the other as an SNR (object Sp70). However, a careful inspection of the top and bottom panels in Figure 18 reveals that Sp70 has two components which are separated by 055; this only becomes apparent by blinking the images centered on data cube layers 348 and 350. The short spectra extracted with a 3 pixel radius aperture for CX22 (top) and CX26 (bottom) are shown on the left-hand side of Figure 18 (explanation as in Figure 5).

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One can ask whether the probability of PN superpositions depends strictly on the underlying surface brightness of the galaxy. Certainly the evidence from narrowband studies supports this hypothesis (e.g., Ciardullo et al. 1989; Jacoby et al. 1990; McMillan et al. 1993), but observations through ∼50 Å wide bandpasses are not nearly as effective as MUSE at surveying the bright inner regions of galaxies. If the number density of PNe does follow the light, it would mean that the effect of blends on the PNLF is highest near the nucleus and increases with the distance of a given galaxy. Figure 19 shows the distribution of blends as a function of galactocentric radius, which in turn is linked to the continuum surface brightness of the galaxy via de Vaucouleur's law. The number density indeed is correlated with the surface brightness and exhibits a steep rise toward the nucleus. For radii smaller than 5'', the PNLF is becoming incomplete, so no further increase is observed. The number statistics is too poor to allow for a more detailed investigation; however, the trend is clear.

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Measurements of the [O iii] λ5007 flux from two blended objects with a line separation of 3.75 Å cannot easily be performed with simple aperture photometry; it requires careful PSF extraction in data cubes using software similar to that used by Kamann et al. (2013) for crowded stellar fields. We have as yet not attempted to adapt this technique to the challenging problem of faint emission-line objects. Instead, we employed an interactive line-fitting tool that allows us to trim the contaminating spectral line with an ad hoc assumption about the true line profile for the object in the center of the aperture. This approach is not rigorously objective but is an improvement over the poor fits that were produced before deblending (see Figure 18).

For the current work, we identified objects with multiple emission-line components by blinking the images between the relevant data cube layers. This procedure allowed us to associate the correct components with their corresponding spatial images. Ideally one would like to automate the process, but we defer that discussion to a later paper. For now, our method of visually blinking the frames has enabled us to identify blends and improve both the photometry and line-of-sight velocity measurements.

Table 3 summarizes our final catalog of confirmed PNe in NGC 1380. In total, we identify 118 PNe in the CENTER field, 40 in the MIDDLE field, and 8 in the HALO field. For comparison, Sp2020 found 91 PNe in the CENTER field of NGC 1380, with a significant fraction (15 objects, or 16% of their total) identified in our survey as blends (see Appendix B, Table 5). We also detected 70 PNe candidates near our detection limit that we classify as unconfirmed, as they are visible only in [O iii] 5007 Å, i.e., they have no other confirming emission line. Of 264 point-source candidates (PN and other) identified by visual inspection through 15 data cube layers, we classify only 16 (6%) as spurious, i.e., their S/N was too low for validation. These numbers demonstrate that our differential imaging approach to PN identification is much more efficient at finding objects near the detection limit and unraveling blended point sources than techniques that work solely with the original data cubes.

Table 4. Emission-line Point Sources Detected in NGC 628

Pointing	Confirmed PNe	Candidate PNe	SNRs	H ii	Seeing (1)	Weather (2)	Δm(3)
K1	16	3	10	28	077	clear	+0.10
K2	31	5	20	18	083	clear	−0.20
P1	11	...	16	7	076	clear	−0.10
P2	9	3	12	11	114	clouds	−0.03
P3a	13	3	19	39	149	clear	−0.40
P3b	13	3	19	39	095	clouds	−0.40
P4	11	...	9	16	108	clear	−0.11
P5	10	...	15	35	105	clouds	−0.20
P6	31	5	17	21	069	clouds	−0.00
P7	38	12	17	37	070	clouds	−0.35
P8	13	2	6	28	082	clouds !	−0.80
P9	6	3	13	23	096	clouds !	−0.95
P12	15	4	6	32	075	clouds !	−0.14

Note. (1) Seeing FWHM (ESO Archive information). (2) Retrieved from ESO Archive: "clouds": some clouds registered during the night, "clouds !": clouds passing during an exposure, (3) zero-point offset used to match the He2008 photometry.

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More information on the classification procedure appears in Section 4.1.4 below.

We performed DAOPHOT aperture photometry on the objects found in all three NGC 1380 pointings. Aperture corrections derived from stars in the field are tabulated in Appendix B, Table 6, and the results of the photometry are given in Appendix C, Table 9. A cross-reference to the identifications of Sp2020 is provided in Column 2. These data allow us to perform a detailed comparison of magnitudes, S/N estimates, radial velocities, and object classifications of the two data sets.

Sp2020 do not provide error estimates for their m₅₀₀₇ photometry. However, we can make a meaningful comparison of the photometric uncertainties using A/rN, the ratio of the fitted emission-line amplitude to the residual continuum noise for the [O iii]λ 5007 emission line. Figure 20 shows our A/rN values versus those quoted by Sp2020. As our off image includes measurements of the background continuum at the position of each PN, we can track the behavior of A/rN versus the underlying galaxy surface brightness. This information is color-coded into the diagram, with blue representing objects projected onto regions of low surface brightness, and red displaying objects superposed on a bright background. Regardless of PN magnitude, and except for a single outlier, the S/N for the Sp2020 data is below the 1:1 line, and typically only ∼60% of that obtained from DAOPHOT aperture photometry on DELF-filtered images. This result is a direct confirmation of the expected advantage of the differential filtering approach as outlined in Section 3.2. Sp2020 excluded any objects from their analysis that fall below a threshold of A/rN = 3. In our DELF photometry, only one of those objects is close to this threshold, and only 3 are below a level of A/rN = 5. In contrast, Sp2020 reported 50 objects below this latter value, again supporting the expectation of a significant gain from our approach.

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Figure 21 compares our m₅₀₀₇ magnitudes to those of Sp2020 using the same color coding as in Figure 10. Although the scatter between the measurements is larger than that expected from our internal tests (see Figure 12), the relation generally follows the 1:1 line. Moreover, a closer inspection of the residuals reveals a trend in the data: objects superposed on regions of higher galaxy background surface brightness (red symbols) tend to be brighter in the Sp2020 data (mean residual of −0.23 mag), and the opposite is true for PN projected onto regions of lower surface brightness (blue, mean residual of +0.14 mag). In the absence of Sp2020 m₅₀₀₇ error estimates, we have used the reciprocal of the quoted A/rN values as a proxy for their error bars in both plots, with the caveat that they are likely underestimates (see Figure 10). Note that we confirm within the error bars the overluminous object reported by Sp2020.

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More instructive is a comparison with the observations of Feldmeier et al. (2007, hereafter Fm2007), who used a narrowband filter on the Magellan Clay telescope to identify 44 PN candidates in a 5.5 arcmin² region north of NGC 1380's nucleus. Sp2020 compared their photometry to this data set and reported good agreement in a relative sense for 4 matching objects in the CENTER field and 17 matches in the MIDDLE field. However, they reported that magnitudes measured by MUSE were systematically 0.45 mag fainter than those of Fm2007. Although we were unable to compare their magnitudes as they only published the photometry for the CENTER field, we decided to repeat the exercise, finding 4 matches in the CENTER, 18 matches in the MIDDLE, and 4 additional matches in the HALO. These matches include Fm2007 objects Fm1 and Fm3, which are present on both the MUSE CENTER and MIDDLE pointings, and Fm29, which is located on both the MIDDLE and HALO fields. We also discovered that thanks to the overlap of the CENTER, MIDDLE, and HALO fields, there are nine PNe common to the CENTER and MIDDLE, and four objects common to the MIDDLE and HALO. A comparison of the pairs of magnitudes reveals that a satisfactory agreement is reached with an offset of −0.4 magnitudes for MIDDLE and HALO with respect to CENTER, suggesting a possible systematic error in the MUSE flux calibration, perhaps caused by nonphotometric conditions. If true, this would essentially reconcile the 0.45 mag discrepancy with Fm2007 as reported by Sp2020.

Comparison plots for each field can be found in Appendix A, Figure 33, while the combined data points for all fields are shown in the right panel of Figure 21. The best agreement with the 1:1 line is achieved assuming a zero-point offset of −0.1 mag for the CENTER field and −0.4 mag for the MIDDLE and HALO pointings, roughly in line with the differential correction described above. However, while there is generally good agreement with the 1:1 line, a number of outliers are apparent. A careful inspection of the stack of diff images reveals that the outliers Fm1, Fm7, and Fm21 (magenta points that fall below the 1:1 line) can be explained by the contamination of PNe light by emission from diffuse gas. This effect is revealed by the velocity separation of the two components in the MUSE spectra. Fm13, which also has an anomalously bright narrowband magnitude, is similarly identified as the blend of two overlapping point sources. The discrepancy for objects Fm19 and Fm33 (the red objects above the 1:1 line) is likely due to the velocities, as their [O iii] λ5007 emission lines (5034.48 and 5032.14 Å) lie on the blue edge of the narrowband filter's bandpass. Just a ∼20% change in the filter transmission would explain the magnitude difference seen in the figure. The final discrepant object, Fm37, is located at the very edge of a MUSE data cube and thus may have unreliable photometry.

Based on these data, we conclude that except for the above outliers, the agreement between Fm2007 and our photometry is good. In the absence of a proper calibrator, the −0.4 mag offset could either be due to a systematic error either in the flux calibration or the aperture correction (or both). To avoid this issue, future targeted observations must be sure to have sufficiently bright PSF stars in the field and have flux standard exposures specifically attached to the observations.

Spectra for all detected PN candidates were obtained using the entire MUSE data cube as described in Section 3.4. The main objective of this exercise was to confirm that the point-like [O iii] emitters are true PNe and to exclude interlopers such as H ii regions, SNRs, and background galaxies. As a byproduct of this step, radial velocities were measured and tabulated for future use in kinematic analyses.

For a candidate to be classified as a PN, it was required to exhibit at least two emission lines, normally [O iii]λ 5007 and [O iii]λ 4959, with the latter's flux measured to be of roughly one third of the former (Storey & Zeippen 2000). For some faint objects near the detection limit, [O iii]λ 4959 was not visible; but Hα was. When Hα was visible, we used the relationship for bright planetaries found by Herrmann et al. (2008, hereafter He2008) and classified the object as a PN if the flux in Hα was smaller than the flux in [O iii] λ5007. If the [O iii] λ5007 line was the only line detected, the object was classified as a PN "candidate." Most of these candidates should be true PNe: while single line detections could be due to background objects such as [O ii] galaxies and Lyα emitters (LAEs), [O ii] emitters would likely be detected in the continuum (Ciardullo et al. 2013), while LAEs are relatively rare. Specifically, at the depth and redshift window of the NGC 1380 data, the surface density of LAEs is roughly 0.5 objects per MUSE pointing (Herenz et al. 2019). Moreover, while the density of LAE contaminants will increase with depth, most Lyα emitters have line widths that are significantly wider than that expected from the [O iii] line of a planetary (e.g., Trainor et al. 2015; Verhamme et al. 2018; Muzahid et al. 2020). Nevertheless, to be conservative, single-line PN candidates were not included in our PNLF analysis.

Some of the NGC 1380 PN candidates that have bright Hα also have significant emission in the low-ionization lines of [N ii] λλ6548, 6584 and [S ii] λλ6717, 6731. This is generally the signature of shock ionization from an SNR. However, NGC 1380 does not exhibit strong star formation activity, nor does it contain much cold interstellar medium, so it is unclear whether these spectral features should be attributed to SNRs. Moreover, our generalized DELF processing about Hα reveals a ∼2 kpc diameter gas disk around the galaxy's nucleus. This disk, which has a kinematic structure similar to that seen for the PNe, has been investigated previously with the GMOS IFU (Ricci et al. 2014), and more recently with MUSE and ALMA data (Tsukui et al. 2020). As illustrated in Figure 22, the disk consists of a combination of diffuse gas and filaments with [N ii]/Hα and [S ii]/Hα line strengths indicative of shock excitation. A number of our point-like and sometimes not quite point-like objects have similar line ratios, suggesting they may actually be physically related to the disk, rather than the result of local supernova explosions. In any case, these strong [N ii] and [S ii] emitters are not planetary nebulae. Although the disk is interesting on its own right, its investigation is beyond the scope of this paper.

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We note that the agreement between the object classifications of Sp2020 and those of this work is generally good. The only differences are for SP40, which Sp2020 classified as an interloper, but we show it to have a PN-like spectrum; Sp22 (CX45), which we classify as a PN but Sp2020 lists as an SNR; and Sp72 (CX115), which we consider a PN, but Sp2020 classifies as an H ii region. For the rest of the Sp2020 objects our classifications agree.

The left panel of Figure 23 compares the luminosity function of objects securely classified as PNe in the three fields of NGC 1380 with that derived from NGC 1380's CENTRAL field by Sp2020. The diagram contains several features of note.

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First, the central field of NGC 1380 contains one PN that is "overluminous," i.e., it appears significantly brighter than the value of M* predicted from the rest of the PN population. Like Sp2020, we have eliminated this object from our analysis, as its inclusion would greatly worsen the overall fit to the empirical function (decreasing its likelihood by a factor of ∼3 × 10⁵). Still, the object presents a puzzle: its spectrum looks like that of an ordinary PN, with a very high [O iii]/Hβ ratio, negligibly faint lines of [N ii] and [S ii], and an [O iii]/Hα ratio consistent with that seen in other bright planetaries (Herrmann et al. 2008). Because the object is only ∼0.25 mag brighter than M*, its apparent luminosity could be explained by a superposition of two PNe within the top ∼0.5 mag of the luminosity function. However, there is no evidence for two components in the shape of the [O iii] λ5007 emission line. Specifically, we used the line-fitting tool pPXF (Cappellari & Emsellem 2004; Cappellari 2017) to measure the [O iii] λ5007 emission line more accurately than is possible by our Gaussian fitting algorithm. We find that the line profile is indistinguishable from the instrumental profile and there is no evidence of doubling. If the object is composed of two separate sources, their positions and velocities must be consistent to within roughly 04 and 75 km s⁻¹ (one spectral bin), respectively.

Alternatively, if the overluminous source identified in the MUSE observations of NGC 1380 is a planetary nebula, it might be foreground to the galaxy. Fornax is known to have a substantial population of intracluster stars, (e.g., Spiniello et al. 2018; Cantiello et al. 2020; Spavone et al. 2020), and because NGC 1380 sits securely in the cluster core, 06 from the central cD galaxy NGC 1399, it is reasonable to assume some of these stars will be in the foreground. In fact, several examples of apparently overly luminous PNe have been found in the Virgo Cluster (Jacoby et al. 1990), another system that is known to have a large population of intracluster stars (e.g., Williams et al. 2007; Longobardi et al. 2015; Mihos et al. 2017). If this is the explanation for the apparent brightness of the PN, then the object is at least ∼2 Mpc in front of NGC 1380, near the turnaround radius of the cluster (Drinkwater et al. 2001). Although the presence of an intracluster M* PN at the extreme edge of the galaxy cluster may seem unlikely, if the PN were much closer to the galaxy, it would not be identified as an outlier in the system's luminosity function. Thus, the hypothesis cannot be ruled out.

With the current data, it is difficult to know whether either of these explanations is correct. But it is important to recognize that sources that appear too bright for the empirical function given by Equation (10) are occasionally found in PN surveys of other galaxies. Also, it is interesting to note that numerical PNLF simulations (e.g., Mendez & Soffner 1997; Valenzuela et al. 2019) do show a shallower slope at the bright end and can produce overluminous PNe when the sample size is large. Méndez et al. (2001) have demonstrated such an effect for a sample of 535 PNe detected in NGC 4697 (see their Figures 14 and 15). Unless the PN sample contains enough objects to reliably define the shape of the luminosity function, or unless further research develops a theory for the existence of these objects, this source of contamination can lead to systematically lower PNLF distance estimates.

Figure 23 also vividly illustrates the advantage of using DELF images rather than the normal MUSE data cubes. Our luminosity function monotonically increases to m₅₀₀₇ ∼ 27.8 before signs of incompleteness begin to set in. The Sp2020 data set also reaches this limit, but only in the outer regions of the CENTER field, where the surface brightness is relatively low. Identifying faint and even intermediate-brightness PNe in regions of high background is difficult without first subtracting a continuum; this is reflected in the Sp2020 sample.

Finally, as the right-hand panel of Figure 23 illustrates, despite coming from the same data, the distance we derive from our PN photometry is slightly less (∼0.07 mag) than the one obtained from the Sp2020 data set. The cause of this difference can be inferred from Figure 21. Because the empirical PNLF has a sharp exponential cutoff, PNLF distances depend most strongly on the observed magnitudes of the brightest few PNe in the sample. Figure 21 demonstrates that in NGC 1380, these bright PNe lie primarily in the lower surface brightness regions of the galaxy, and for these objects, the Sp2020 magnitudes are systematically fainter than our measurements by a few hundredths of a magnitude. This translates into the offset seen in Figure 23. The probability distributions also show the effect that our smaller measurement uncertainties and a larger sample size have on the likelihood distribution: the internal errors associated with our sample are ∼60% smaller than those computed from the Sp2020 data set.

The maximum-likelihood solutions shown in Figure 23 bring up an interesting issue for future PNLF measurements. If we assume a foreground reddening of E(B − V) = 0.046 (Schlafly & Finkbeiner 2011) and an absolute value for M* = −4.53 (Ciardullo 2013), then the most likely distance modulus to NGC 1380 is ${\left(m-M\right)}_{0}\,=\,{31.10}_{-0.05}^{+0.04}$ (16.6 Mpc), with additional systematic uncertainties associated with the errors on the frames' aperture corrections, flux calibration, and the amount of foreground reddening. For comparison, the SBF method generally produces values between 31.20 and 31.60 (e.g., Tonry et al. 2001; Blakeslee et al. 2009, 2010). This offset of ∼0.3 mag is not unexpected and may be explained by a very small amount of internal reddening in the Cepheid galaxies used for calibration. As has been pointed out by Ciardullo et al. (1993) and again by Ciardullo et al. (2002a), both the PNLF and SBF methods use relatively late-type galaxies to define the zero points of their distance scales. However, any systematic difference between the amount of internal reddening present in these galaxies and that within the early-type systems targeted by the methods will lead to the SBF scale being overestimated and the PNLF scale being underestimated, with Δμ ∼ 7ΔE(B − V), i.e., ΔE(B − V) ∼ 0.04.

Alternatively, we can compare our distance modulus to those derived from Cepheid and TRGB measurements of other cluster galaxies. NGC 1326A and 1365 are both projected within ∼1 Mpc of the Fornax central cD galaxy (NGC 1399), have radial velocities consistent with cluster membership, and have been surveyed for Cepheids by the HST. Their mean Cepheid-based distance modulus of (m − M)₀ = 31.10 (Freedman et al. 2001) is identical to our measurement. On the other hand, the TRGB distance to NGC 1365 is 0.19 mag more distant than our value (Jang et al. 2018). Moreover, if NGC 1425 (a galaxy projected 56 away on the sky) is also considered to be part of Fornax, then its Cepheid distance ((m − M)₀ = 31.60; Freedman et al. 2001), when averaged with those of NGC 1326A and 1365, makes the cluster's mean Cepheid distance consistent with that of the TRGB. So it is possible that the PNLF distances are still biased slightly toward smaller values.

Except for the data reduction, where we immediately used the fully reduced data cubes as downloaded from the ESO archive, the basic steps for analysis were identical to the ones described for NGC 1380 in Section 4.1. Source detection was facilitated on the one hand by the fact that the narrow velocity dispersion perpendicular to the disk reduced the number of data cube layers that needed to be inspected. However, the large number of point-like H ii regions and SNRs complicated the issue. We were able to efficiently identify a promising sample of PNe by supplementing the detections of the DAOPHOT FIND algorithm with visual inspection of the data cubes. We then performed photometry and spectroscopy on the source list to measure emission-line strengths and classify the objects via their line ratios (see Table 4).

[O iii] maps for all fields are presented in Appendix A, Figure 35. As was found in the analysis of NGC 1380, careful inspection of the images was needed to identify blends. Figure 25 shows a few examples of multiplicity, some of which have escaped detection in previous studies. Such cases can explain the [O iii] magnitude differences between our MUSE data and the measurements of other studies (see Section 4.2.2 below).

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[O iii] measurements were performed in the standard way as described above. However, our photometry was limited by two issues. The first was weather: according to the ESO archive, most of the MUSE observations of NGC 628 were affected by clouds. More serious was the lack of point sources in the field. Unlike NGC 1380, we found it difficult to identify foreground stars suitable for aperture correction measurements. As shown in Figure 6, the FWHM versus wavelength relation for NGC 1380 was well behaved and followed the expectations of atmospheric turbulence. However, in NGC 628, there were significant differences between the point-source candidates of each field, and for some objects, we even measured the FWHM to increase with wavelength. Such an effect could conceivably be produced if the target objects were not actually stars, but globular clusters with a large population of blue stragglers concentrated in the core. Attempts to create a model PSF by stacking PN images also proved problematic, as the result was sensitive to the surface brightness of the galaxy's ubiquitous diffuse background emission. The aperture corrections, which are the dominant source of uncertainty in our photometry, and the catalog of PN candidates are presented in Appendices B and C, Tables 7 and 10, respectively.

Because of these difficulties, we used the He2008 PN photometry as the flux standard system for our measurements. A comparison of our corrected m₅₀₀₇ values with the He2008 data is documented in Appendix A, Figure 36. The agreement between the two data sets is generally good, except for the outliers caused by blending. A collapsed version of measurements for all 51 objects in common with He2008 is shown in the left panel of Figure 26. For PNe brighter than m₅₀₀₇ = 27.0 the residuals display a standard deviation of 0.12 mag, which is in reasonable agreement with the uncertainties quoted by He2008 (±0.058 mag at m₅₀₀₇ = 26.0 and ±0.135 mag at m₅₀₀₇ = 27.0).

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The right panel of Figure 26 adjusts our photometry in fields K1, K2, and P2 by −0.2 mag and compares the measurements with those of Kr2017. The plot is broken down by object classes, with H ii regions coded in orange, SNRs in red, and PNe in green. The magenta points at the bright end of the distribution are objects where undetected source confusion affected the Kr2017 photometry. The standard deviation for the PN measurements is 0.21 mag, in agreement with the Kr2017 error estimates. See also Figure 34 in Appendix A for comparison plots per field K1, K2, and P1.

Figure 27 shows a comparison of the m₅₀₀₇ magnitudes for all emission-line point sources (PNe, H ii regions, SNRs) obtained in field P3 at two different observing epochs under different observing conditions. While the He2008 photometry had a completeness limit of m₅₀₀₇ = 26.5, the MUSE data extends roughly 2.5 magnitudes fainter, with an internal dispersion of 0.15 mag and a mean residual of 0.004 mag for PNe brighter than 28.0 mag (excluding outliers). The bright outliers (magenta) between 25 < m₅₀₀₇ < 27 are H ii regions and SNRs in crowded regions that suffer from blending (several examples are shown in the right panel of Figure 25). For this internal consistency check, no zero-point shift has been applied.

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The left-hand panel of Figure 28 displays the bright end of the PNLF for NGC 628. Although our PN observations extend all the way to m₅₀₀₇ ∼ 29, we only use the top ∼1.5 mag of the function in our fit. This is due to the fact that, in star-forming populations, the PNLF is not monotonic: instead of exponentially increasing at faint magnitudes, the function exhibits a distinctive dip two to four magnitudes below M*. This feature, first identified by Jacoby & De Marco (2002), occurs in PN populations that are dominated by objects with intermediate- and high-mass cores and is discussed in Ciardullo (2010). Because Equation (10) does not attempt to model this downturn, we truncate our fit at m₅₀₀₇ ∼ 26.9 and do not use the faintest ∼55% of our sample.

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The right-hand panel of Figure 28 displays the uncertainty in our distance modulus. Assuming a foreground galactic extinction E(B − V) = 0.062 (Schlafly & Finkbeiner 2011), our most likely distance modulus is ${\left(m-M\right)}_{0}\,=\,{29.76}_{-0.05}^{+0.03}$ (or 8.95 Mpc), where the uncertainties represent only the statistical errors of the fit. When analyzed using our maximum-likelihood technique and using the same value of M*, the Kr2017 data give ${\left(m-M\right)}_{0}\,=\,{29.76}_{-0.05}^{+0.12}$, while the PNe of He2008 yield ${\left(m-M\right)}_{0}\,=\,{29.73}_{-0.07}^{+0.06}$. However, we should point out that this apparent consistency hides a potentially important issue: the brightest PN in our sample (P7-38), though not strictly overluminous, is 0.11 mag brighter than the next brightest source. There is no reason to exclude the object from the analysis, as our best-fit to Equation (10) is clearly acceptable. However, if do we exclude the object, the best-fit solution becomes ∼90 times more likely, and the galaxy's distance increases to ${29.87}_{-0.05}^{+0.03}$.

This issue points out an important limitation of using planetary nebulae for distance determinations. To determine a robust distance using the PNLF, one cannot depend solely on the magnitude of the most luminous PN in a galaxy. Instead, one has to define the shape of the brightest ∼1 mag of the PN luminosity function. Unfortunately, PNe in this critical magnitude range are rare: the specific PN densities given by Ciardullo et al. (2005), coupled with the galaxy bolometric corrections computed by Buzzoni et al. (2006) imply that an M_V = −19 galaxy will only contain ∼20 PNe in the magnitude range of interest. In the era of precision cosmology, this number is not sufficiently constraining, as the distance may be susceptible to contamination by bright interlopers, or it may be underpopulated, leading to a large uncertainty in distance. As Figure 28 illustrates, a minimum of ∼50 objects are needed for a robust measurement of the PNLF shape. PN surveys must therefore sample at least M_V ∼ −20 of a galaxy's luminosity.

The data retrieval and emission-line analysis proceeded as above. The fully reduced data product as available in the ESO archive consists of a single cube, where two data sets of total exposure time 5.1 hr were merged into one. Therefore, in the overlapping region of the two fields shown in Figure 29, the total exposure time was more than 10 hr. As can be seen in the off-band image displayed in Figure 30(a), the faint surface brightness in this ring region is strongly modulated by the residual flat-field pattern discussed in Section 3.2. This systematic error has completely vanished after DELF processing, yielding the diff images shown in Figure 30(b). The exquisite quality and depth of this region allowed us to easily confirm the eight objects found by Fe2020 and identify seven additional PNe candidates. (Because three of these new objects are single-line detections, we classify them as possible PNe and do not use them in our analysis.) Thus, the observations yielded 12 likely PNe with [O iii] magnitudes in the range 28.5 ≲ m₅₀₀₇ ≲ 30.2. Serendipitously, our PN search also discovered two z = 4.551 Lyα emitters, one of which overlaps with PN8. The region containing PN8 and a Lyα emitter is shown in the inset of Figure 30.

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As was our experience with NGC 1380, the selection of point sources for measuring the cube's PSF and aperture correction was nontrivial, as many of the brighter objects are either globular clusters or background galaxies. Using three objects marked in Figure 30(a), we measured [O iii] FWHM values of 076 in the combined field and 078 in the southern field. Precision photometry would require us to accurately measure the PSF in each subfield separately, but for the sake of simplicity in this experiment, we have ignored the (very small) difference in image quality. The adopted aperture corrections and our catalog of PN candidates are found in Appendices B and C, Tables 8 and 11, respectively.

To compare our photometry with the literature, we needed to convert the absolute M₅₀₀₇ values listed by Fe2020 to apparent m₅₀₀₇ magnitudes. However, with their assumed distance modulus of (m − M)₀ = 32.45, the comparison resulted in an offset of 0.5 mag. With a choice of (m − M)₀ = 32.95 and a foreground extinction of 0.1 mag (Schlafly & Finkbeiner 2011), we obtain excellent agreement as shown in Figure 31. The single outlier at m₅₀₀₇ = 28.9 is probably due to the presence of three continuum point sources very close to the centroid of our PN4 (PN1 in the nomenclature of Fe2020).

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Note that the gray plot symbols, although self-referenced to our own magnitudes, are only intended to illustrate the range of photometry possible with deep MUSE exposures. These data, which should be compared with the simulations described in Section 3.6.2 and shown in Figure 12, demonstrate that a ∼10 hr exposure with MUSE can reach [O iii]λ 5007 magnitudes as faint as m₅₀₀₇ ∼ 31. Moreover, with the MUSE image quality that is currently being achieved with adaptive optics this limit can be improved significantly. For example, in the MUSE Extremely Deep Field combined data cube, an image quality of 06 FWHM at 4700 Å has been achieved over a total exposure time of 140 hr (Bacon et al. 2021). Thus, it should be possible to push PN detections well past the distance of NGC 474.

Figure 32 illustrates our PN luminosity function for NGC 474. The PN sample is small, not because of any issue with the quality or depth of the MUSE observations, but because the amount of galaxy luminosity encompassed by the survey fields is small. As a result, the data cannot be used to obtain a robust measure of the distance to the galaxy. Nevertheless, if we assume that all the PNe are drawn from the empirical function of Equation (10), we can perform a maximum-likelihood analysis on the data and obtain an estimate of the galaxy's distance. We adopt a 90% completeness magnitude limit of m₅₀₀₇ = 29.1; this ad hoc assumption could be improved by running a series of artificial star experiments on the data cube, but is sufficient for our purpose. If we then assume a foreground E(B − V) = 0.03 (Schlafly & Finkbeiner 2011), the most likely distance modulus obtained from the PNe is ${\left(m-M\right)}_{0}\,=\,{32.86}_{-0.25}^{+0.08}$, or ${37.4}_{-4.0}^{+1.5}$ Mpc. Again, these error bars only represent the formal uncertainties of the fits. Systematic errors associated with the aperture correction, foreground reddening determination, and the assumed value of M* are not included in the calculation. This distance is fully consistent with that obtained from the SBF analysis (Cantiello et al. 2007).

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