ABUNDANCES OF GALACTIC ANTICENTER PLANETARY NEBULAE AND THE OXYGEN ABUNDANCE GRADIENT IN THE GALACTIC DISK^*

Observations of 37 PNe were carried out on the ARC 3.5 m telescope at Apache Point Observatory, NM, using the Dual Imaging Spectrograph (DIS). Each side of the DIS has an E2V 2048 × 1024 pixel CCD. The DIS allows simultaneous observations to be made in the blue and red portions of the spectrum, providing the important advantages of identical slit placement and identical sky conditions for the entire optical-near-infrared spectrum from 3600 to 9600 Å.

We observed with a 360'' by 2'' slit generally oriented E–W. Counts on the CCD were binned by 2 pixels (08) along the slit direction before readout. The seeing was generally 1''–15, though for this program of extended objects seeing has little impact except possibly for the flux calibration star. On the blue side, we used the B400 grating giving 1.83 Å pixel⁻¹, and a resolution of ∼7 Å. On the red side, we used the R300 grating, yielding 2.31 Å pixel⁻¹, for a resolution of ∼9 Å. The spectroscopic data are noisy between 5400 and 5700 Å where the red and blue spectra overlap; in addition, the sensitivity of the CCD decreases at λ < 3900 Å and >8500 Å.

Observations were obtained during five runs in 2007 between January and November. We reduced the data in the standard fashion with IRAF.⁹ Bias and overscan levels were subtracted and a normalized flat field was applied. We checked the data for response variations in the spatial direction along the slit and found that there was no need to apply an illumination correction. For each PN, we extracted an appropriate spatial portion of the two-dimensional spectrum, collapsing it to one dimension, taking particular care to extract the same size spatial swath in the blue and the red, since the spatial scales differ between the blue (0.40 arcsec pixel⁻¹) and the red (0.42 arcsec pixel⁻¹) CCDs. The wavelength scale was derived from observations of arc lamps, and the flux calibration from observations of two or three standard stars each night. Table 1 shows our observing log.

We also report here new results for four additional PNe included in the gradient analysis in Section 3. These objects, observed in 2003 and 2004 with the Kitt Peak National Observatory 2.1 m telescope or the Cerro Tololo Inter-American Observatory (CTIO) 1.5 m telescope, are listed at the bottom of Table 1. See Section 2 in Milingo et al. (2010) for details on the instrumental configurations and observing procedures. These data were also analyzed in the standard fashion using IRAF, as outlined above.

For each PN, the extracted spectra were combined to make a single blue spectrum and a single red spectrum. The only exceptions arose when a strong line was saturated in the normal-length exposures: [O iii] λ5007 in the blue and Hα in the red. In those cases, we obtained an additional blue and/or red short-exposure spectrum used to measure just the line that was saturated in the longer exposures.

We measured emission-line fluxes from the final combined, flux-calibrated blue and red spectra of each PN. These fluxes formed the input for our abundance determinations. We used the ELSA package (Johnson et al. 2006) for the subsequent plasma diagnostics and abundance calculations. The first step in the analysis is to generate a table of line intensities that have been corrected for interstellar reddening and for contamination of the hydrogen Balmer lines by coincident recombination lines of He⁺⁺. The reddening curves that we employed were taken from Seaton (1979, UV), Savage & Mathis (1979, visual), and Fluks et al. (1994, IR). ELSA iterates calculations of the density (from [S ii] λ6717/λ6731) and temperature (from [O iii] λ5007/λ4363) until a convergent solution is reached, then calculates the amount of contamination to subtract. It also calculates the correct Hα/Hβ ratio for the converged values of temperature and density. Table 2 lists the observed fluxes and corrected intensities, including their uncertainties, along with the calculated value of c (the logarithmic reddening parameter), the appropriate value of Hα/Hβ, and the observed Hβ flux through the spectrograph slit.

Table 2. Fluxes and Intensities

Line	f(λ)	A 2		A 8		A 12		A 14
		F(λ)c	I(λ)	F(λ)	I(λ)	F(λ)	I(λ)	F(λ)	I(λ)
[O ii] λ3727	0.292	79.2	116 ± 26	230	416 ± 93	273	382 ± 86	180	326 ± 73
Ne iii λ3869	0.252	76.6	106 ± 23	60.8	102 ± 21	82.3	110 ± 23	38.1 ::	63.5 ± 33.86::
He i + H8 λ3889	0.247	13.8	19.0 ± 4.00	12.5	20.7 ± 4.31	13.9	18.4 ± 3.86	...	...
Ne iii λ3968	0.225	24.0a	32.0 ± 9.80a	23.8a	37.5 ± 10.59a	25.4a	32.9 ± 9.90a	...	...
H λ3970	0.224	12.2a	16.3a	9.58a	15.1a	12.5a	16.2a	...	...
He i + He ii λ4026	0.209	2.04::	2.68 ± 1.42::	...	...	...	...	...	...
He ii λ4100	0.188	0.452a	0.577a	0.144a	0.211a	0.249a	0.309a	...	...
Hδ λ4101	0.188	20.3a	25.9 ± 5.04a	25.1a	36.8 ± 7.04a	19.5a	24.2 ± 4.65a	...	...
He ii λ4339	0.124	0.869a	1.02a	0.295a	0.380a	0.474a	0.546a	0.416a	0.536a
Hγ λ4340	0.124	40.7a	47.8 ± 8.36a	32.6a	42.0 ± 7.25a	42.0a	48.5 ± 8.41a	47.2a	60.7 ± 10.50a
⋮
S iii λ9069	−0.670	92.2	38.8 ± 8.90	102	26.2 ± 5.96	38.9	18.0 ± 4.13	22.6:	5.84 ± 2.12:
P9 λ9228	−0.610	9.70	4.41 ± 0.93	...	...	4.78	2.37 ± 0.50	...	...
S iii λ9532	−0.632	188	83.0 ± 18.01	256	71.1 ± 15.30	108	52.4 ± 11.38	56.5:	15.8 ± 5.61:
P8 λ9546	−0.633	8.35	3.68 ± 0.80	...	...	6.47	3.12 ± 0.68	...	...
C i λ9824	−0.653	...	...	...	...	11.9:	5.62 ± 2.03:	...	...
C i λ9850	−0.654	...	...	...	...	38.6	18.2 ± 4.08	...	...
c		0.56		0.88		0.50		0.88
Hα/Hβ		2.83		2.86		2.84		2.86
log FHβb		−13.59		−14.21		−12.42		−14.82

Notes. ^aDeblended. ^berg cm⁻² s⁻¹ in our extracted spectra. ^cThe F(λ) values have been scaled to Hβ = 100 using the observed F_Hβ values. Intensities of strong lines have measurement uncertainties of 10%, single colons indicate uncertainties of 25%, double colons indicate uncertainties 50%, and triple colons indicate highly suspect measurements.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Along with the [O iii] temperature and [S ii] density calculated as described above, ELSA calculates the [N ii] temperature from λ6584/λ5755. If the required lines are observed, the [S iii] temperature from λ9532/λ6312 or λ9069/λ6312, the [S ii] temperature from λλ6717+6731/λλ4068+4076, the [O ii] temperature from λ3727/λ7323, and the [Cl iii] density from λ5517/λ5537 are also calculated. For most of the ionic abundance determinations to be described below, either the [O iii] or [N ii] temperature was used along with the [S ii] density.

Ionic abundances for our sample are given in Table 3. The table shows the emission line used to calculate each listed value of ionic abundance along with the diagnostic temperature used in the calculation. The designation "wm" refers to the weighted mean of the ionic abundances, weighted by the observed flux of the line used to calculate it. Only lines flagged with an asterisk are included in the weighted mean. At the end of each element's list of ionic abundances in Table 3 is the ICF. This factor represents an empirical attempt to correct the observed ionic abundances for contributions by unobserved ionization states. The ICFs themselves take advantage of similarities in ionization potentials among different elements, though for some, like sulfur, there are no close similarities, and we rely on model predictions. See, e.g., Kwitter & Henry (2001) for a complete discussion of the methods used to compute ionic abundances and the ICFs used in our work.

Electron densities and temperatures for each object are presented in Table 4, while final elemental abundances and selected element-to-element ratios are given in Table 5. The statistical uncertainties provided in all cases are the result of careful propagation by ELSA of line strength measurement errors, while systematic uncertainties are not included.

In the Appendix, we compare our oxygen abundances with others from the literature. We assert that although the abundance agreement is generally very good, there are some outliers.

Table 6 is a compilation of information for our sample of 124 PNe that is used in the analysis to follow. For each object identified in Column 1, Column 2 provides the heliocentric distances taken from Cahn et al. (1992) for each of the objects. All but eight of these values were available in Cahn et al. (1992). References for the remaining eight objects are provided in a table footnote. The corresponding galactocentric distances in Column 3 were computed using the formula given in a footnote to Table 6, where we took the Sun's galactocentric distance to be 8.5 kpc. Employing a distance method that was calibrated using PNe in the LMC, Stanghellini et al. (2008, hereafter SSV) provide updated heliocentric distances for 101 of our 124 objects. These values and their corresponding galactocentric distances are listed in Columns 4 and 5. Finally, Column 6 lists the final oxygen abundances from Table 5, using the customary form 12+log(O/H). The following analysis is based largely on the Cahn et al. (1992) distances, but we use the SSV results to demonstrate the effects of using different distance scales.

Figure 1 is a plot of 12+log(O/H) versus galactocentric distance (Table 6, Column 3) in kpc for the 124 objects in our PN sample with well-measured distances. The PN sample is a compilation of objects from Henry et al. (2004), Milingo et al. (2010), plus those objects reported upon for the first time in this paper (HK10). Our sample contains only those objects classified as either Type I or Type II, i.e., objects whose location and kinematics indicate that their progenitors were members of the disk population. We use the heliocentric distances from Cahn et al. (1992) from which we calculate galactocentric distances, except for the following objects (heliocentric distance sources in parentheses): H2-18, He2-48, He2-55 (Maciel 1984); IC 418 (Guzmán et al. 2009); M3-15 (Zhang 1995); NGC 6720, NGC 6853, NGC 7293 (Harris 2006); H3-75 (Amnuel et al. 1984); K3-64, K3-93 (Jaskot 2008); and St3-1 (Tajitsu & Tamura 1998).

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Our primary goals are (1) to subject the data to careful statistical tests in order to determine the nature of the oxygen abundance gradient in the Galactic disk as measured by PNe and (2) to compare our result with analogous studies using other types of probes. In order to be included in the gradient analysis, we required that a PN have good determinations of distance, plus good T_e[O iii], T_e[N ii], and N_e[S ii], which means reliable flux measurements of λ4363, λ5755, and λλ6717/6731. Combining the 41 PNe whose measurements are presented here with additional disk PNe already in our database, 124 objects qualified for inclusion in the sample on which we performed a thorough linear regression analysis.

Our first step was to gauge the strength of a correlation between O and R_g by computing a Pearson correlation coefficient, r, for the entire sample of objects, using the program pearsn in Press et al. (2003). The Pearson correlation coefficient is the ratio of the covariance of two variables—in this case O and R_g—to the product of their individual standard deviations. The square of this coefficient, r², called the coefficient of determination, is a measure of the strength of the linear relationship between the two variables. Both r and r² are invariant under linear transformation, and thus uncertainties in O and R_g are irrelevant. (See Rodgers & Nicewander (1988) and Hays (1981) for a more detailed description of the correlation coefficient.)

For our complete PN sample, we found that r = −0.54 and thus r² = 0.29. The proper interpretation of this result is that 29% of the variability in 12+log(O/H) (henceforth O) is accounted for by R_g under the assumption that a linear model exists which describes the O–R_g relation. The same statistical program also predicts the probability that this result could arise from a completely uncorrelated parent population. Thus, if the null hypothesis is that O and R_g are uncorrelated in the parent population, then the probability that a sample of 124 objects drawn from such a parent population will have a value of |r|⩾ 0.54 is only 3.3 × 10⁻¹¹. In other words, there is a vanishingly small probability that this null hypothesis is being falsely rejected (Type I error; Hays 1981), and so the correlation appears to be truly nonzero.

Our next step was to derive a good linear model, i.e., least-squares fit, for the data in the O–R_g plane. We did this by using the program fitexy in Press et al. (2003), which accounts for errors in both coordinates: O and R_g. Our initial attempt included the 1σ uncertainties in O taken directly from our published estimates, while for R_g we assumed a standard 1σ uncertainty of ±20% (Stanghellini et al. 2006). These factors resulted in a slope, b, of −0.066 ± 0.0055 dex kpc⁻¹ and an intercept, a, of 9.15 ± 0.04. Assuming that the errors are normally (Gaussian) distributed, this fit has an associated χ² value of 178.0 with 122 degrees of freedom ν (124 sample objects minus 2 determined parameters, a and b) or a reduced χ² (χ²_ν = χ²/ν) of 1.46.

Ideally, we would like the value of χ²_ν to be unity; values greater than this suggest that the random errors have been underestimated (or there is an unidentified factor that results in the scatter being greater than what is described by the uncertainties). Meanwhile, values less than unity indicate that one has overestimated the errors. In the spirit of Goldilocks, when χ²_ν ≈ 1, then the assumed uncertainties satisfactorily explain the scatter of the data points around the model. A method of quantifying this idea is to compute the probability that a value of χ²_ν greater than or equal to the one determined from the data could result from randomly sampling the parent population described by the model and for the same number of degrees of freedom (Bevington & Robinson 2003, Chap. 11). This is done by integrating the χ² probability function from χ²_ν to infinity to obtain ${q}_{\chi ^2}$, often referred to as a goodness-of-fit parameter. A relatively large ${q}_{\chi ^2}$ suggests that within the confines of the 1σ errors, χ²_ν determined from the data has a reasonable chance of occurring and the model is likely adequate in characterizing the data. Likewise, a relatively small value means that the errors cannot explain χ²_ν and that the model is likely inadequate. The threshold value for ${q}_{\chi ^2}$ in statistical studies is usually taken to be 0.05 (Hays 1981). In our current case, $q_{\chi ^2}$ was determined by the program fitexy to be 0.00074, and so we deemed this particular trial model unsatisfactory.

Before continuing further with our analysis, we decided to write our own program for computing linear least-square fits in order to verify the accuracy of the routines in Press et al. (2003) as well as to understand the methods better. Therefore, we wrote a program in Mathematica to generate a least-squares fit of data with errors in both coordinates. Specifically, we solved iteratively the exact "least-squares cubic" derived by York (1966, 1969) as unified by York et al. (2004) (see also MacDonald & Thompson 1992). To verify our procedures and implementation and assess the accuracy of the resulting fitting parameters, we successfully checked two test cases: (1) the classic data case developed by Pearson (1901) with errors assigned to both coordinates by York (1966, Table II) and previously considered by Lybanon (1984, 1985), Powell & Macdonald (1972), Reed (1992), and York et al. (2004) and (2) data for electron temperatures of ionized hydrogen in the Magellanic Clouds derived by Vermeij & van der Hulst (2002) from abundances of O⁺² and S⁺² and previously considered by York et al. (2004).

In attempting to produce a fit to our own data sample of 124 objects, we assumed the same 1σ errors as in the above trial with fitexy. Only six iterations were required to determine the slope and intercept to 10 decimal places; the rounded values are b = −0.066 ± 0.006 and a = 9.15 ± 0.04, a χ² value of 178.0 and a χ²_ν of 1.46. The calculated standard deviations in the slope and intercept (the "standard errors") were computed following the guidelines in Cowan (1998, Chap. 9). Since our results are in excellent agreement with those produced by the routines in Press et al. (2003), we confidently continued our analysis using the latter formulations.

We sought to improve the fit by scaling the uncertainties of both O and R_g upward independently, since increasing the standard deviation of either or both coordinates reduces the size of χ²; correspondingly, the value of ${q}_{\chi ^2}$ goes up. For each PN the new uncertainty σ_O = σ'_O × f(O), where σ'_O is the originally determined uncertainty in O and f(O) is the scaling factor. For R_g, we set $\sigma _{{R_g}} = {R_g} \times {f(R_g)}$. Thus, we experimented with different values of the scaling factors f(O) and f(R_g) in systematic fashion in order to find reasonable ranges in these numbers for satisfactory fits. Values of f(O) ranged between 1 and 2, while values of f(R_g) ranged between 0.0 and 0.3.

The test results for each combination of scaling factors are given in Table 7, where the first two columns list the scaling factors, the third column lists the resulting slope of the fit, and the fourth column provides the value for $q_{\chi ^2}$, with the rows ordered top to bottom by increasing $q_{\chi ^2}$. We see that acceptable models correspond to gradient slopes ranging between −0.041 and −0.074 dex kpc⁻¹. Additional tests not reported here indicated that larger values of f(R_g) will push the lower limit downward but it is not likely to go below −0.08. However, realistic values of f(R_g) likely do not greatly exceed 0.3, based upon the brief discussion in Stanghellini et al. (2006).

Table 7. Trial Least-square Fits

f(O)	f(Rg)	b	${q}_{\chi ^{\rm 2}}$
1	0	−0.041	1.95E-28
1	0.1	−0.051	4.70E-16
1.25	0	−0.041	3.77E-10
1.25	0.1	−0.049	4.92E-06
1	0.2	−0.066	0.00071
1.5	0	−0.041	0.0030
1.5	0.1	−0.047	0.050
1.25	0.2	−0.061	0.13
1.75	0	−0.041	0.42
1.75	0.1	−0.045	0.71
1.5	0.2	−0.056	0.75
1	0.3	−0.074	0.78
2	0	−0.041	0.96
1.25	0.3	−0.070	0.98
2	0.1	−0.045	0.99
1.75	0.2	−0.053	0.99
2	0.2	−0.051	1
1.5	0.3	−0.066	1
2	0.3	−0.059	1
1.75	0.3	−0.062	1

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To refine our model further, we assumed that f(R_g) = 0.2, consistent with the value recommended by Stanghellini et al. (2006), and then adjusted the value of f(O) until the fit explained all of the scatter in O, i.e., χ²_ν = 1.0. This criterion was met when f(O) = 1.40, at which point the resulting model had the following parameters: a = 9.09 ± 0.05 dex, b = −0.058 ± 0.006 dex pc⁻², χ² = 121.8, χ²_ν = 1.00, and ${q}_{\chi ^2}=0.49$. Since the last statistic is well above the threshold of 0.05, the model cannot be rejected, and we judge the fit to be acceptable. These results for the total PN sample are shown in the first row of Table 8, where the first column describes the sample or subsample under consideration, the second column provides the number of objects in the sample, and the next seven columns give the y intercept, slope, χ², reduced χ²_ν, goodness-of-fit parameter, correlation coefficient, and the correlation probability factor, respectively. (Subsequent rows provide the statistics for PN subsamples or other object types.)

Table 8. Oxygen Gradients

Samplea	n	a	b	χ2	χ2ν	$q_{\chi ^2}$	r	pr
Total	124	9.09 ± 0.05	−0.058 ± 0.006	121.8	1.00	0.49	−0.54	3.3(−11)
Type I	48	9.10 ± 0.07	−0.061 ± 0.008	62.7	1.36	0.051	−0.59	3.5(−6)
Type II	75	9.05 ± 0.08	−0.053 ± 0.010	58.6	0.80	0.89	−0.44	4.8(−5)
R < 10 kpc	85	9.04 ± 0.09	−0.054 ± 0.013	73.3	0.88	0.77	−0.21	4.8(−2)
R > 10 kpc	38	9.92 ± 10.10	−0.12 ± 0.14	24.2	0.67	0.93	−0.39	1.2(−2)
H ii regions	73	9.24 ± 0.07	−0.078 ± 0.007	63.5	0.89	0.72	−0.80	1.2(−20)
B V stars	24	9.63 ± 0.11	−0.080 ± 0.01	33.3	1.51	0.058	−0.64	2.6(−4)

Note. ^aThe first five groups refer to the PN sample presented here, where the abundance uncertainty for each object was increased by 40% and the error in the galactocentric distance was assumed to be 20%. The H ii region sample was assembled from studies by Afflerbach et al. (1997), Vílchez & Esteban (1996), and Deharveng et al. (2000), and the B V star sample was compiled from the papers by Smartt et al. (2001) and Rolleston et al. (2000).

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Thus, we adopt the following analytical behavior of the oxygen abundance with galactocentric distance (Table 6, Columns 3 and 6) as that which best describes the data for the total sample presented in Figure 1:

$$\begin{equation} 12+\log({\rm O/H}) = (9.09\pm 0.05) -(0.058\pm 0.006) \times {{\it R}_{\it g}}. \end{equation} \tag{ 1 }$$

This regression model is indicated with a solid bold line. At the Sun's distance from the Galactic center, 8.5 kpc, our model predicts a value of 8.60 ± 0.07 for 12+log(O/H), close to a recently determined solar value of 8.69 ± 0.05 by Asplund et al. (2009).

We checked to make sure that our inferred gradient in Equation (1) was not being influenced too heavily by the few extreme points at large R_g. To do this, we started with the object with the greatest galactocentric distance and excluded PNe one by one, calculating a new value for the gradient each time. All resulting gradients obtained in this manner were slightly flatter—the maximum slope was −0.051, occurring after eliminating all of the outer six objects. Thus, we feel confident that the model in Equation (1) is not being influenced unduly by the few objects in our sample located at large radial distances.

The need to increase the value of σ'_O by 40% beyond our previously established level of uncertainty in order to reach χ²_ν = 1 suggests that objects in our PN sample possess some natural scatter which is related to real abundance differences among them. Such differences could be caused by one or more of the following: (1) age differences of the progenitor stars, where the older ones formed out of less metal-rich material than younger ones; (2) inhomogeneous mixing of the interstellar material out of which the progenitors formed; (3) stellar diffusion, in which stars migrate along the disk from a galactocentric radius where they formed to their present location; and/or (4) systematic errors in the abundance determination process. It is clear from Table 7 that the value of the gradient is sensitive to uncertainties in R_g as well as to the oxygen abundance. However, our adopted standard uncertainty of 0.20 for R_g is considered to be quite reasonable (L. Stanghellini 2010, private communication), and thus we still maintain that at least some natural scatter in the oxygen abundances exists. In any case, it is beyond the scope of this paper to further discuss the origin of the natural scatter.

For comparison purposes and to see the effects of using a different set of distances, we used the 101 objects in Table 6 for which SSV provided heliocentric measurements in order to repeat the regression statistics computation. We first compared directly the two distance sets and found that while most objects fell very close to a line of 1:1 correspondence, roughly 10% of the objects did not. Interestingly, these differences turned out to have a noticeable impact on the regression results when performed with the SSV distances. Using the same uncertainties that produced the result in Equation (1), i.e., f(O) = 1.4 and f(R_g) = 0.2, the SSV distances now imply a slope of −0.042 ± 0.004 with χ²_ν = 1.40. Additional tests involving larger (probably unrealistic) values of f(R_g) to drive χ²_ν down resulted in a further flattening of the slope. We shall comment on this result at the end of this subsection.

Next, we subdivided our sample by galactocentric distance, Peimbert type, and height above the Galactic plane. In the first two cases, we computed a least-squares fit and correlation coefficient for each subgroup to look for distribution differences among them.

We divided objects into two groups based upon whether their galactocentric distance placed them outside or inside the 10 kpc circle. This distance was chosen because it coincides with the position of the sharp discontinuity in [Fe/H] reported by Twarog et al. (1997) in which the mean value of [Fe/H] beyond this point was found to drop by 0.3 dex. We then computed the regression statistics separately for the two groups. PNe out to 10 kpc have a gradient and correlation coefficient of −0.054 ± 0.013 and −0.21, while those beyond this distance have values of −0.12 ± 0.14 and −0.39, respectively, and their curves are shown in Figure 1. In the same figure, we also show a quadratic fit to the data which also indicates a steepening of the gradient, where the function corresponding to this curve is 12 + log(O/H) = 8.81 − 0.014 × R_g − 0.0011 × R_g². The quadratic fit predicts a relatively flat gradient at small galactocentric distances but a rather steep one past 10 kpc. Although these results suggest that the gradient steepens in the outer disk, we caution that the scatter is broad beyond 10 kpc, and only additional data at these distances will allow us to reliably determine the true behavior of the slope. Interestingly, some authors present empirical evidence for a flattened gradient in the outer disk (Vílchez & Esteban 1996; Costa et al. 2004; Maciel & Costa 2009; Pedicelli et al. 2009). For example, Costa et al. (2004) use PNe to estimate that the oxygen gradient is −0.09 dex kpc⁻¹ between 4 and 5 kpc, but the slope becomes flat at 11 kpc and beyond. Likewise, Pedicelli et al. (2009), using a sample of 265 Cepheids between 5 and 17 kpc from the Galactic center, find the iron gradient to be three times steeper inside the 8 kpc circle than outside of it. This disagreement is also apparent in published model results. All of the chemical evolution models by Fu et al. (2009) and the models employing a constant star formation efficiency by Marcon-Uchida et al. (2010) predict a steepening gradient in the outer disk, while the models by Marcon-Uchida et al. (2010) which assume that the star formation efficiency falls off with radial distance predict a flattening of the gradient.

Figure 2 shows the distributions by Peimbert type along with the regression lines, while the complete statistical information is provided in Table 8. The gradient slopes (and uncertainties) were found to be −0.061 ± 0.008 and −0.053 ± 0.010 for Type I and Type II, respectively, while the correlation coefficients were −0.59 and −0.44, respectively. Thus, within the uncertainties the two types appear to be indistinguishable, a conclusion supported visually in Figure 2. In addition, the intercept values are also comparable, and so we are unable to see any statistically significant difference in oxygen abundance patterns between the two PN types. This conflicts with the claim of a gradient difference between PN types by S10, who find a slope of −0.035 ± 0.024 dex kpc⁻¹ for Type I PNe and a slope of −0.023 ± 0.005 dex kpc⁻¹ for Type II PNe. These authors continue by relating PN type with progenitor age and conclude that the gradient has steepened with time. Interestingly, their conclusion conflicts with that of Maciel & Costa (2009), who claim that the gradient has flattened over time. Resolving the problem of temporal slope change is extremely important from the standpoint of chemical evolution of the Milky Way Galaxy (MWG), but more than likely the situation is currently being complicated by small-number statistics along with real scatter in the oxygen abundances of objects at similar galactocentric distances.

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Figure 3 separates the sample by vertical distance from the Galactic plane, where we adopt Z = 300 pc, the general disk population scale height (Cox 2000). The gradient slopes were found to be −0.038 ± 0.007 and −0.033 ± 0.009 for objects above and below this height, respectively, while the correlation coefficients are −0.63 and −0.39, respectively. These numbers, along with a visual inspection of the lines in the figure, indicate that the distributions of the two subsamples are indistinguishable.

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We repeated the above exercise but this time adopted Z = 100 pc, the Population I scale height (Cox 2000). The regression analysis inferred slopes of −0.038 ± 0.006 and −0.013 ± 0.028 for PNe above and below 100 pc, respectively, while the correlation coefficients are −0.59 and −0.09, respectively. Here the differences, particularly in the correlation coefficients, are likely the result of the relatively small galactocentric distance range for those objects within 100 pc of the plane.

Finally, Equation (1) was used to normalize all oxygen abundances to the same galactocentric distance and then these adjusted oxygen abundances were plotted against the corresponding z distance perpendicular to the Galactic plane. While we do not show the plot here, we can report that no correlation was found. We conclude that within our database there is no evidence for a correlation between oxygen abundance and vertical distance from the plane.

Table 8 also provides comparisons of our derived PN oxygen abundance gradient for the disk of the MWG with analogous results for what are likely to be younger populations of H ii regions and B V stars. The bottom two rows of Table 8 show the results of applying our statistical methods to an H ii region and a B V star sample which we compiled using abundances taken directly from the literature. The H ii region sample was compiled from measurements reported in Afflerbach et al. (1997), Vílchez & Esteban (1996), and Deharveng et al. (2000), and covers the galactocentric distance range of 3–17 kpc. We used the abundance uncertainties as quoted in the papers and adopted a standard error in galactocentric distance of 2 kpc for an acceptable fit. Likewise, the B V star sample was compiled from studies by Smartt et al. (2001) and Rolleston et al. (2000) and extends from 2 to 18 kpc. In this case, it was necessary to scale the abundance uncertainties by 1.9 in order to achieve a suitable fit, while the galactocentric distance errors were taken directly from the papers. We present a comparison of the total PN, H ii region, and B V star samples listed in Table 8, along with their corresponding least-square fits, in Figure 4.

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In both the H ii region and stellar studies, we see strong inverse correlations between oxygen abundance and galactocentric distance and good agreement in slope values with that of our total PN sample. Note that the correlation coefficients in both cases indicate the presence of a stronger correlation than for our total PN sample, although the gradients are consistent. Interestingly, this is despite the fact that there is likely to be a mean age difference between our PNe and the H ii region and B V samples, as the latter two generally represent a younger population than the first group.

Table 9 presents a broad comparison of our adopted gradient (Equation (1)) for our total sample with many other gradients from PN studies along with H ii region, Cepheid, and B V star studies. In contrast to the gradients listed in Table 8, which we computed directly from the published data, the gradients in Table 9 are those provided by the authors. For the source identified in Column 1, the second column gives the number of objects in the sample, while the next three columns provide the gradient in dex kpc⁻¹, the type of objects in the sample, and the galactocentric distance range covered in kpc.

First, we see a wide range of gradient values among the PN samples listed. In particular, the difference between this paper and S10 is interesting, since both samples contain a large number of objects and cover a broad range in galactocentric distance. Clearly, the gradients inferred by these two studies are statistically different, with the S10 paper proposing a flatter gradient than the one we find here. We shall expand on the meaning of this difference in the next subsection. In the meantime, we verified that the difference in the linear models derived by S10 and us was not attributable to the use of different oxygen ICFs. We recomputed our abundances using their ICF formula for oxygen, taken from Kingsburgh & Barlow (1994), and found essentially no difference in either slope or intercept.

The other three PN-derived gradients by Henry et al. (2004, hereafter H04), Costa et al. (2004, hereafter C04), and Perinotto & Morbidelli (2006, hereafter P06) cover slightly more restricted distance ranges. H04 used their own spectrophotometric observations to derive abundances of Types I and II PNe in the MWG disk; C04 likewise used their own spectral measurements to derive abundances of 26 Galactic anticenter disk objects and combined these with PNe from the sample of Maciel & Quireza (1999) to infer their gradient; and P06 compiled observations from the literature of Type II PNe but reprocessed all line strengths themselves to produce a homogeneous abundance set for determining their gradient. The gradients of S10, H04, and P06 appear to be very similar, given the uncertainties associated with each.

The remaining gradients in Table 9 include those derived from optical and IR studies of H ii regions by Rudolph et al. (2006, hereafter R06), Cepheids by Pedicelli et al. (2009, hereafter P09), and B V stars by Smartt & Rolleston (1997, hereafter SR97). (Note that P09 report an Fe/H gradient.) Gradients in these four cases are consistent with those for PNe, although the uncertainties are somewhat larger in the former.

In summary, we see strong evidence once again for a negative oxygen gradient in the disk of the MWG. However, there is no evidence in our own work that the slope differs between PN Types I and II or between objects of different vertical distance from the Galactic plane. Because of the broad abundance spread beyond 10 kpc, we cannot confirm with any confidence the claim by many others that the gradient changes (either steepens or flattens) in the outer regions of the disk. We therefore adopt the simplest hypothesis, to wit, for now that the gradient is constant along the disk's entire length.

The most troubling aspect of our gradient derivation exercise is that both the uncertainties in the distance scale and the oxygen abundances prevent us from pinning down the value of the slope at this time to an accuracy better than about 0.02 dex kpc⁻¹. This became apparent above when we substituted the SSV distances for the ones used in the original analysis, i.e., Column 2 of Table 6, and were forced to add more natural scatter to the oxygen abundances in order to reduce the value of χ²_ν to an acceptable level. From our complete exercise, we consider it very likely that the true slope is within the range of −0.04 to −0.06 dex kpc⁻¹, but we cannot refine the number beyond that point. Essentially, we have reached a confusion limit regarding the abundance gradient as derived using PNe. Perhaps with the launch of the Gaia probe¹⁰ in 2012 trigonometric distances to many Galactic PNe can be measured with much better accuracy, and the uncertainty introduced by the distance scale for PNe can be significantly reduced.

Given the range in values for the slope of the Galactic disk oxygen gradient displayed in Table 9, the obvious uncertainties related to an unsettled distance scale, and the extensive work that is currently going on to reconcile these differences, it is appropriate to ask whether continuing our efforts to better define the characteristics of the oxygen abundance distribution is likely to offer much in the way of additional improvement in our understanding of the chemical evolution of the MWG disk. To address this question, we consider the model-predicted sensitivity of the Galactic oxygen gradient to a few parameters related to galactic chemical evolution.

Recent detailed chemical evolution models of the Milky Way disk by Marcon-Uchida et al. (2010) illustrate how the value of the present-day gradient as reflected by studies of H ii regions is influenced by the threshold density for star formation, i.e., the surface density above which star formation occurs but below which it does not, and the variation of the star formation efficiency, i.e., the ratio of the mass of stars formed to the mass of gas available for forming them per unit time. Their models predict that between 4 and 14 kpc in galactocentric distance, when the star formation efficiency is held constant and the star formation threshold is reduced from 7 to 4 M_☉ pc⁻², the predicted oxygen gradient flattens, going from −0.059 to −0.025 dex kpc⁻¹. Note that these gradient values are almost exactly the same as those reported in this paper (−0.058) and the one by S10 (−0.023), respectively, for our total samples. At the same time, holding the threshold constant at 4 M_☉ pc⁻² but allowing the star formation efficiency to be a function of galactocentric distance produces little effect. Similar patterns are seen over more restricted ranges in R_g. Therefore, these models indicate that improving our knowledge of the oxygen gradient in the Galactic disk will allow us to pin down the value of at least one important parameter, the star formation threshold.

In a similar fashion, detailed chemical evolution models of the Milky Way disk by Fu et al. (2009) tested the effects of the infall timescale, the star formation law, and disk formation timescale (DFT) on the size of the abundance gradient. In their models, the disk is assumed to begin forming out of matter infalling from the halo at a time specified by the DFT, while the rate of infall decreases by e⁻¹ over a time defined by the infall timescale. At the same time, the star formation rate is proportional to some power of the local gas surface density and may also be a function of radial distance along the disk.

Fu et al. (2009) tested the impact of these three parameters on the steepness of the present-day abundance gradient and found that the DFT had the largest effect, while the other two parameters influenced the gradient in a minor way only. For example, when DFT is zero (disk formation begins at the moment of Galaxy formation) the gradient ranged in value from −0.009 to −0.027 dex kpc⁻¹. However, when the DFT value increases radially the resulting gradient range steepens to −0.056 to −0.091. The range in values in each case is the result of changing the star formation law or the infall timescale. The point here, of course, is that as in the case of the Marcon-Uchida et al. (2010) models we see a predicted range in gradient steepness that corresponds roughly to the observed range in the oxygen gradient presented in Table 9.

Thus, chemical evolution models strongly indicate that some important parameters of chemical evolution of the disk, the density threshold for star formation, and the timescale for disk formation are expected to have a measurable influence on the value of the abundance gradient. Therefore, continuing to refine our quantitative description of the gradient is clearly worthwhile. Further studies should continue to focus on extending the baseline out to a radial distance beyond 20 kpc from the Galactic center in order to firmly establish the character of the gradient in terms of its actual value in addition to its shape, since there is observational support for a gradient of constant slope plus evidence for gradients which steepen or flatten with increasing galactocentric distance.