Chemical Abundances of Planetary Nebulae in the Substructures of M31. II. The Extended Sample and a Comparison Study with the Outer-disk Group

We report deep spectroscopy of 10 planetary nebulae (PNe) in the Andromeda Galaxy (M31) using the 10.4 m Gran Telescopio Canarias (GTC). Our targets reside in different regions of M31, including halo streams and the dwarf satellite M32, and kinematically deviate from the extended disk. The temperature-sensitive [O iii] λ4363 line is observed in all PNe. For four PNe, the GTC spectra extend beyond 1 μm, enabling the explicit detection of the [S iii] λ6312 and λλ9069, 9531 lines and thus determination of the [S iii] temperature. Abundance ratios are derived and generally consistent with AGB model predictions. Our PNe probably all evolved from low-mass (<2 M☉) stars, as analyzed with the most up-to-date post-AGB evolutionary models, and their main-sequence ages are mostly ∼2–5 Gyr. Compared to the underlying, smooth, metal-poor halo of M31, our targets are uniformly metal rich ([O/H] ≳ −0.4), and seem to resemble the younger population in the stream. We thus speculate that our halo PNe formed in the Giant Stream’s progenitor through extended star formation. Alternatively, they might have formed from the same metal-rich gas as did the outer-disk PNe but were displaced into their present locations as a result of galactic interactions. These interpretations are, although speculative, qualitatively in line with the current picture, as inferred from previous wide-field photometric surveys, that M31's halo is the result of complex interactions and merger processes. The behavior of the N/O of the combined sample of the outer-disk and our halo/substructure PNe signifies that hot bottom burning might actually occur at <3 M☉ but careful assessment is needed.


INTRODUCTION
In the cold dark matter (ΛCDM)-dominated universe, large galaxies formed hierarchically (e.g., White 1978;White & Rees 1978) through accretion/merger of smaller subsystems.Such interactions tidally disrupt smaller galaxies and result in extended stellar halo surrounding the central galaxy (e.g., Ibata et al. 2007Ibata et al. , 2014)).The relics of galaxy interaction and assemblage are registered into the extended halo in forms of stellar streams which, if detected, can be used to study the properties of galaxies and backtrack past interactions (e.g., Ibata et al. 2001a,b,c;Ferguson et al. 2002;Majewski et al. 2003;McConnachie et al. 2009).
The Andromeda Galaxy (M31) is a nearby (785 kpc, McConnachie et al. 2005) large spiral system and an ideal candidate for studying galaxy formation and evolution.Wide-field surveys, such as PAndAS 3 , have revealed in M31's outer halo a wealth of large-scale stellar substructures extending to nearly 150 kpc from the galactic centre (e.g., Ibata et al. 2001aIbata et al. , 2007;;Ferguson et al. 2002;McConnachie et al. 2003McConnachie et al. , 2004McConnachie et al. , 2009;;Irwin et al. 2005;Tanaka et al. 2010), with the Northern Spur and the southern Giant Stellar Stream (hereafter the Giant Stream, Ibata et al. 2001a;Caldwell et al. 2010) among the first discovered.The Giant Stream threads to the southeast halo, as far as >4 • from the centre of M31 (Ibata et al. 2001a;McConnachie et al. 2003).-PN18 was discarded from analysis because no nebular emission lines were detected in its spectrum.a Number in the bracket is the ID from Merrett et al. (2006) except PN10, PN13 and PN17, which were discovered and identified in the LAMOST survey (Yuan et al. 2010).
d Sky-projected galactocentric distance estimated at a distance of 785 kpc to M31 (McConnachie et al. 2005).
e Here "Halo" means that the PN belongs to the outer halo, or is associated with some substructure.
f PN nature confirmed by the LAMOST survey (Yuan et al. 2010).
g Might be associated with the NE Shelf, as explained in Section 4.4.
The Northern Spur is a feature with enhanced density in metal-rich red giant branch (RGB) stars, located at ∼2 • towards the north (Ferguson et al. 2002).Planetary nebulae (PNe) are descendants of low-and intermediate-mass (∼1-8 M ) stars, which account for the majority of stellar populations in our universe.Given their bright, narrow emission lines, PNe are excellent tracers of the chemistry, dynamics and stellar populations of their host galaxies.In the optical spectrum of a PN, the bright [O iii] λ5007 nebular line alone can carry ∼10% of the central star's energy (e.g., Schönberner et al. 2007).PNe thus are well detected in distant galaxies, even as far as >100 Mpc (e.g., Gerhard et al. 2005Gerhard et al. , 2007;;Longobardi et al. 2015a,b).Spectroscopy of PNe in M31, mainly in the bulge and disk (e.g., Jacoby & Ciardullo 1999;Richer et al. 1999;Kwitter et al. 2012) has found a slightly negative gradient in the oxygen abundance within 50 kpc in the disk (Kwitter et al. 2012).However, recent observations with large (8-10 m) telescopes found that the outer-disk PNe, as far as 100 kpc from the centre of M31, have nearly solar abundances (Balick et al. 2013;Corradi et al. 2015).Even some of the PNe associated with the substructures have O/H close to the Sun (Fang et al. 2013(Fang et al. , 2015)).These metal-rich PNe in the outskirts of M31 seem to have different origins from the ancient halo, which formed through galaxy mergers long time ago (e.g., Ibata et al. 2007Ibata et al. , 2014)).
One long-standing, unresolved question is what the origin of M31's stellar substructure is.It has been proposed that the Northern Spur and the Giant Stream might be connected by a stellar stream (Ferguson et al. 2002;Merrett et al. 2003), of which the dwarf satellite M32 could be the origin (Ibata et al. 2001a;Merrett et al. 2003), but this hypothesis needs assessment.In pursuit of answering this question, we have carried out deep spectroscopic observations of ten bright PNe associated with the two substructures and mostly located in the outer halo (Fang et al. 2013, 2015, hereafter Papers I andII, respectively), and found that they are in overall metal-rich ([O/H]∼ −0.3-0) and their oxygen abundances are consistent within the errors (although some internal scatter exists).These abundance analyses led to a tempting, yet tentative, conclusion that the Giant Stream and the Northern Spur might have the same origin.Given the vast extension and complexity of M31's halo (Ibata et al. 2007(Ibata et al. , 2014)), our sample of PNe so far observed, although representative, is still too limited for us to draw any definite conclusion.
That both the PNe on the halo streams and those kinematically belonging to the extended disk of M31 have been found to be metal-rich (∼solar) is unexpected for a classical, metal-poor halo, and leads to a question of whether they have the same origin (or even population).A comparison study between our halo sample and the disk objects may shed light on this conundrum.Previous attempts have proved that PNe are a very efficient probe of different regions of M31.It is thus possible not only to assess the connection and origin of different substructures, which has been the motivation of our observations so far, but also to make a census study of the extended halo of M31, using PNe as a tool.
In order to better understand the merging history of M31's halo, we recently carried out deep spectroscopy of eleven PNe: three in the Northern Spur, three associated with the Giant Stream, two in M32, and another three located in the eastern and southeast halo regions.The immediate objectives of the new observations are 1) to obtain accurate abundances (mainly oxygen) for an extended sample, 2) to make a comparison study with the outer-disk PNe in terms of abundance and stellar population, and 3) to assess whether M32 is related to the Northern Spur and the Giant Stream.Being the third paper targeting the PNe in the substructures of M31, this paper is the second one of a series to report deep spectroscopy with a 10 m class telescope.Section 2 introduces target selection and describes the observations and data reduction.Section 3 present emission line measurements, plasma diagnostics and abundance determinations.We present in-depth discussion in Section 4 based on the results, and give summary and conclusions in Section 5.

OBSERVATIONS AND DATA REDUCTION
2.1.Target Selection: The Spatial and Kinematical Distribution Before introducing target selection, we brief some definitions in terms of boundaries in M31 structurre.We adopted the M31 bulge radius (∼3.4 kpc) from the surface brightness fitting by Irwin et al. (2005).The inner disk of M31 is defined at R 25 (=95 , de Vaucouleurs et al. 1991), which corresponds to 21.7 kpc at the distance of M31; this radius well encompasses the optical disk of M31.Beyond R 25 lies the extended disk that stretches to 40 kpc, with detections as far as ∼70 kpc (Ibata et al. 2005).In the current paper, all M31 PNe beyond R 25 but with kinematics consistent with the extended disk are dubbed the outer-disk PNe.Previous spectroscopic observations of Kwitter et al. (2012), Balick et al. (2013) and Corradi et al. (2015) all focused on the outer-disk PNe in M31.
In Papers I and II, we targeted the PNe in the Northern Spur and the Giant Stream substructures.Since our targets kinematically deviate from the extended disk of M31 and are mostly located in the halo, hereafter we call them the halo PNe, to avoid possible confusion with the outerdisk PNe.For the new GTC observations, we selected a sample that covers not only the substructures, but also more extended areas in the M31 system such as the eastern and the southeast halo regions and dwarf satellite M32.The locations/hosts of our targets are given in Table 1, where other properties such as target positions (right ascension-R.A., declination-Decl.),visual magnitudes in [O iii] λ5007 (m λ5007 ), heliocentric velocities v helio (in km s −1 ), angular distances to the centre of M31, and the sky-projected galactocentric distances (in kpc) are also presented.Spatial locations of our targets, including those studied in Papers I and II, are shown in Figure 1.Our halo nebulae are mostly outside R 25 .
We selected eight PNe from the catalog of Merrett et al. (2006) and three from Yuan et al. (2010); the latter was based on a spectroscopic survey at the Large Sky Area Multi-Object Fiber Spectroscopic Telescope4 (LAMOST, Su et al. 1998;Cui et al. 2004Cui et al. , 2010Cui et al. , 2012;;Zhao et al. 2012).These new targets were named PN8-PN18 (see Table 1 and Figure 1), following the target  Merrett et al. (2006), Hurley-Keller et al. (2004), Yuan et al. (2010) and Kniazev et al. (2014).Our GTC targets (including those in Paper II) are highlighted with color-filled circles, which are colorcoded according to their locations (see Table 1).The three Northern Spur PNe from Paper I are indicated by red asterisks.Coordinates ξ and η (given in Table 1) are the M31-based reference frame defined by Huchra et al. (1991).The green dashed ellipse represents the optical disk of M31 with radius R 25 =95 (de Vaucouleurs et al. 1991), assuming an inclination angle of 77.• 7 (de Vaucouleurs  1958) and a position angle of 37. • 7 (Merrett et al. 2006) for the M31 disk.
naming (PN1-PN7) in Paper II.According to their locations in M31, our GTC samples PN1-PN17 are highlighted with different colors in Figure 1, where PN16 and PN18 are too close to each other and visually indistinguishable.The [O iii] brightnesses of the new sample are m λ5007 ∼20.48-21.96,extending down to nearly 1.8 mag from the bright-end cut-off of the planetary nebula luminosity function (PNLF) of M31 (Merrett et al. 2006;Ciardullo et al. 1989;Ciardullo 2010).Yuan et al. (2010) did not assign their newly discovered PNe to any locations (i.e., substructure or the extended disk).We identified the locations of the three LAMOST targets (PN10, PN13 and PN17) according to their kinematics shown in Figure 2, which also presents the distribution of the line-of-sight velocity with respect to the centre of M31, v los , versus distance along the major and minor axes of M31.The kinematics of PN10 obviously deviates from the extended disk of M31 and is somewhat close to the Northern Spur sample identified by Merrett et al. (2006, Figure 32 therein).We thus identified PN10 as a possible Northern Spur object.PN13 visually resides on the southeast (SE) extension of the Giant Stream; PN15 also seems to be on the stream.However, the velocities of these two PNe, although both deviating from the kinematics of the extended disk, are inconsistent with the stellar orbit of Merrett et al. (2003).PN17 is located in the eastern halo, 2. • 05 from the centre of M31.Its velocity differs significantly from the disk, and its location seems to be very close to the NE Shelf (Ferguson et al. 2005).We temporarily assign PN13, PN15  Kwitter et al. (2012, Kwitter12), Balick et al. (2013, Balick13), andCorradi et al. (2015, Corradi15).In the upper-left panel: over-plotted is the stellar orbit proposed by Merrett et al. (2003, thick green curve; reproduced with kind permission of the authors).In the two side panels (bottom and right): a projection of the orbit in line-of-sight velocity with respect to M31, v los , versus distance along the major and minor axes of M31 is superimposed on the PNe data.Velocities of PNe have been corrected for the systemic velocity −306 km s −1 of M31 (Corbelli et al. 2010).The red dashed curve in the bottom panel is the H i rotation curve from Carignan et al. (2006).and PN17 to be the halo nebulae; detailed discussion is in Section 4. The PN nature of PN14 (ID 2507 in Merrett et al. 2006) was confirmed in the LAMOST survey; it might be associated with the Giant Stream.For the other targets selected from Merrett et al. (2006), we adopted their locations identified by the authors (Table 1).

Spectroscopic Observations
Deep spectroscopy of M31 PNe were carried out with the Optical System for Imaging and low-intermediate-Resolution Integrated Spectroscopy (OSIRIS) spectrograph on the 10.4 m Gran Telescopio Canarias (GTC) at Observatorio de El Roque de los Muchachos (ORM, La Palma).These observations were obtained from 2016 September 2 to 2016 September 11 for GTC program No. GTC25-16B (PI: X. Fang) in service mode.The OSIRIS grism R1000B (1000 lines mm −1 ), which covers ∼3630-7850 Å, and a long slit with 1. 0 width were used.The OSIRIS detector is a combination of two 2048×4096 CCDs.The pixel size is 15 µm, corresponding to 0. 127 in angular size.We adopted the standard observing mode where the output images were binned by 2×2.The above instrument setup produces a spectral resolution of ∼5.5 Å (full width at half-maximum, FWHM) in the blue part of the spectrum and 6.4 Å in the red, at a dispersion of ∼2.072 Å pixel −1 .The ideal observing conditions at the ORM provided photometric and clear nights, and excellent seeing (0. 6-0.8) for most of the observations.Moon was also close to dark during the observations.Throughout the observations, the long slit was placed along the parallactic angles to minimize light loss due to atmospheric diffraction.The typical physical sizes of PNe are 0.5 pc (e.g., Frew et al. 2016), corresponding to 0. 13 in angular size at the distance of M31.This is smaller than the binned CCD pixel size (0. 254) of OSIRIS, and thus our targets are all point sources and supposed to be well accommodated within the GTC 1 -wide long slit.
In order to remove cosmic rays and to avoid saturation of strong emission lines, multiple exposures were made for each target PN.These exposures are summarized in Table 1.In total, 30 hours observations were completed at the GTC for eleven targets.Thanks to the large lightcollecting area of the GTC, we could clearly see almost all the PNe in the direct acquisition CCD image with an exposure of a few seconds (e.g., Figure 3), and then placed the GTC long slit on the targets.Blind offset was utilized only for the two PNe in M32 due to their close proximity (∼7. 4 and 15. 4) to the centre of M32.Exposures of spectrophotometric standard stars Ross 640 and G191-B2B (Oke 1974(Oke , 1990) ) were made in each night to calibrate fluxes for the target spectra, using a slit width of 2. 52.The HgAr and neon arc line images were obtained (with both 1. 0 and 2. 52 slit widths) for wavelength calibration and geometric rectification.Other basic calibration files, such as bias and spectral flats, were also obtained for both the target PN and the standard spectrophotometric star on each night.
We also obtained long-slit spectroscopy of four PNe (PN9, PN11, PN13 and PN17) using the GTC OSIRIS red grism R1000R that covers ∼5080-10370 Å.These observations were obtained on 2016 August 23-24 for program No. GTC66-16A (PI: X. Fang).Slit width was 1. 0, and spectral resolution FWHM∼6.7 Å in the blue region and 8.5 Å in the red, with a dispersion of 2.59 Å pixel −1 .The R1000R exposures are summarized in Table 1.The HgAr, neon and xenon arc lines were used for wavelength calibration.Spectrophotometric standard stars for flux calibration were the same as in the R1000B spectroscopy.Data were obtained under photometric conditions, with seeing ∼0.8-1.0.

Data Reduction
The GTC OSIRIS long-slit spectra were reduced using iraf 5 v2.16.Data reduction generally followed the standard procedure, similar to what has been described in Fang et al. (2015).The raw PN spectral images were first bias-subtracted and corrected for flat-field.We then performed wavelength calibration using HgAr arc lines for the PN spectra obtained with the R1000B grism, and HgAr+Xe for the R1000R spectra.Although geometry distortion along the long slit does not affect the nebular emission lines of our targets, which are point sources on CCD, such distortion of the sky lines must be corrected for so that background subtraction can be properly done.During the wavelength calibration, we rectified geometry distortion by fitting the arc lines using third-order polynomial functions in the two-dimensional (2D) spectrogram.This geometry rectification "straightened" the sky lines along the slit.
We subtracted the background from each single exposure of the target frame by fitting the background emission along the slit direction using high-order cubic spline functions (see more details in Fang et al. 2015).We then combined the background-subtracted 2D frames of the same PN to remove the cosmic rays.We then used the FILTER/COSMIC task in the software midas6 v13SEPpl1.2to further eliminate any possible cosmic residuals in the CCD images.The above procedures produced a well "cleaned" spectral image for each PN, which was then flux-calibrated (and also corrected for the atmospheric extinction) using the spectrum of spectrophotometric standards.
We extracted a 1D spectrum on the fully calibrated 2D frame of each PN for spectral analysis.As an example, Figure 4 shows the 1D spectra for PN12 and PN17 in our sample.In the common wavelength region (5080-7850 Å) covered by the R1000B and R1000R grisms, differences in the fluxes of emission lines (He i λλ5876,6678,7065, [N ii] λλ6548,6583, Hα, [S ii] λ6716,6731, [Ar iii] λ7136, [O ii] λλ7320,7330) detected in both spectra of PN9, PN11, PN13 and PN17 are mostly less than 5%.We corrected for the effect of second-order contamination in the red part of the R1000B spectrum (Figure4, top) following the method of Fang et al. (2015).For the R1000R grism, the second-order contamination exists beyond 9200 Å ( -GTC OSIRIS spectrum of PN12 (top) and PN17 (bottom).The spectrum of PN12 was obtained using the R1000B grism, while the spectrum of PN17 is a combination of the R1000B (≤7500 Å) and R1000R (>7500 Å) spectra.Vertical ranges of both panels are set to accommodate the intensity of Hβ.All important emission lines are labeled.Extinction has not been corrected for.For PN12, the weak features between [O ii] λλ7320,7330 and [Ar iii] λ7751 are the second-order effect.In the spectrum of PN17, the strong emission features redward of [S iii] λ9531 are also due to the second-order contamination.0.94 0.96±0.17
d Blended with [S ii] λ4076; probably also blended with the weak O ii M10 3p 4 D o -3d 4 F and C iii M16 4f 3 F o -5g 3 G lines.
e Corrected for the flux from the blended He ii line.
f Blended with the N iii λλ4634,4642 lines; probably also blended with O ii M1 λλ4639,4642.
g Blended with the O ii M2 3s 4 P-3p 4 D o lines.
h Corrected for the flux from the blended He i λ4713 (2p 3 P o -4s 3 S) line.
k Corrected for the flux from the blended He ii λ6683 (5g 2 G-13h 2 H o ) line.
l Flux underestimated due to the second-order contamination beyond 9200 Å.
m In units of erg cm −2 s −1 , as measured in the extracted spectrum.
lines in one (target PN18) of the two PNe in M32 failed due to its close proximity (7.4) to the bright nucleus of M32, although this target is the brightest in our sample.The other M32 PN (PN16) is 15. 4 from M32's centre and has good data quality.We thus analyzed ten PNe (PN8-PN17; Table 1) in this paper.

RESULTS AND ANALYSIS
3.1.Emission Line Fluxes The emission line fluxes were measured from the extracted 1D spectra by integrating over line profiles.The observed line fluxes of all targets, normalized to F (Hβ)=100, are presented in Table 2, where the observed Hβ fluxes (in erg cm −2 s −1 ) are also presented.The R1000R spectrum was scaled according to the Hα line flux in the R1000B spectrum.We derived the logarithmic extinction parameter, c(Hβ), by comparing the observed and theoretical ratios of hydrogen Balmer lines, Hα/Hβ and Hγ/Hβ.The theoretical Case B H i line ratios were adopted from Storey & Hummer (1995) at an electron temperature of 10 000 K and a density of 10 4 cm −3 .The c(Hβ) values of our PNe are small (0.08-0.25) and are presented in Table 2.The observed line fluxes were then dereddened using the formula where f (λ) is the extinction curve of Cardelli et al. (1989) with a total-to-selective extinction ratio R V = 3.1.The extinction corrected line intensities, all normalized to I(Hβ) = 100, along with the measurement errors, are presented in Table 2. Given the excellent observing conditions (seeing<1.0) and the slit width (1.0), light loss in strong emission lines is expected to be negligible.

Plasma Diagnostics
We carried out plasma diagnostics of PNe using the ratios of the extinction-corrected fluxes of the collisionally excited lines (CELs; also often called forbidden lines) of heavy elements in Table 2.The [S ii] λ6716/λ6731 ratio is a common density diagnostic.Where available, intensity ratio of the fainter [Ar iv] λλ4711,4740 lines was also used to derive the electron density; here the flux of the blended He i λ4713 line was corrected for using the theoretical He i line ratios calculated by Porter et al. (2012).The electron temperature was derived from the [O iii] (λ4959+λ5007)/λ4363 nebular-to-auroral line ratio.The [N ii] temperature was also determined whenever the [N ii] λ5755 line was detected.References for the atomic data utilized in plasma diagnostics as well as the ionic-abundance determinations (in Section 3.3) are summarized in Table 3, where sources of the effective recombination coefficients for the optical recombination lines (ORLs) analyzed in the paper are also given.Results of plasma diagnostics are presented in Table 4.
Only in two PNe (PN12 and PN13) did we find that T e ([N ii]) is reasonably lower than T e ([O iii]).In the other targets (except PN16), T e ([N ii])> T e ([O iii]).This might be due to high-density clumps in PNe (Morisset 2016): at high densities (N e 10 5 cm −3 ), emission of the [N ii] λλ6548,6583 nebular lines can be suppressed due to collisional deexcitation (while emission of the λ5755 auroral line is unaffected), and consequently the [N ii] temperature is overestimated.Results of plasma diagnostics  Howarth & Adams (1981) to solve the statistical equilibrium equations of multi-level atoms to derive level populations and line emissivities under given nebular physical conditions, was used for the plasma diagnostics.
Other temperature-sensitive ratios are [O ii] λ3727/(λ7320+λ7330) and [S ii] (λ6716+λ6731)/λ4072, where λ3727 is a blend of the [O ii] λλ3726,3729 doublet, and λ4072 is a blend of [S ii] λλ4068,4076.However, not in all PNe did we detect these faint auroral lines to a desired S/N.For the four PNe for which the OSIRIS R1000R spectroscopy was obtained, we also derived the temperature using the [S iii] (λ9069+λ9531)/λ6312 line ratio.Since the [S iii] λ9531 nebular line was affected by the second-order contamination (see Section 2.3), we assumed a theoretical ratio λ9531/λ9069 = 2.48 (Mendoza & Zeippen 1982a;Mendoza 1983) to derive the intrinsic flux of this [S iii] line.Besides the traditional CEL diagnostics, we also determined the electron temperatures using the He i ORLs.These He i temperatures were generally lower than those derived from the CELs (Table 4), consistent with Paper II.The principles of PN plasma diagnostics based on the He i ORLs are described in Zhang et al. (2005).
Uncertainties in the electron temperatures and densities presented in Table 4 were estimated based on the measurement errors of emission line fluxes through propagation.Weaker lines generally have larger measurement errors, as a result introducing larger uncertainties in temperatures/densities.A typical example is [N ii] λ5755, whose intensity is 10%-30% that of [O iii] λ4363 for most targets in our sample.Errors in the [N ii] temperatures are systematically higher that those in the [O iii] temperatures, which are thus best measured.

Ionic Abundances
Using the relative intensities of emission lines in Table 2 and the electron temperatures and densities in Table 4, we calculated the ionic abundances of our PNe.The equib program was used to calculate the ionic abun- dances of He, C, N, O, Ne, S, Cl and Ar relative to hydrogen, which are presented in Table 5.The deep GTC spectroscopy enabled detection of several faint diagnostic lines, including the [O iii] λ4363, [N ii] λ5755, and [S iii] λ6312 auroral lines.It is thus possible to consider multiple ionization zones within a nebula, i.e., to assign different temperatures/densities when calculating abundances of ionic species with different ionization stages (i.e., the ionization potentials).This is a more realistic paradigm of nebular analysis, and reduces the uncertainties in resultant abundances that may arise from the temperatures assumed.
The [O iii] temperature was used to derive O 2+ /H + , Ne 2+ /H + , Ar 2+ /H + , Ar 3+ /H + and Cl 2+ /H + .We adopted the [N ii] temperature to calculate N + /H + and O + /H + for PN12 and PN13, where we found the [N ii] temperature is lower than that derived from the [O iii] line ratio.For the other PNe, we considered the recipe of Dufour et al. (2015, also Kwitter & Henry 2001) for the electron temperature in the low-ionization region: if He ii λ4686 was detected, we adopted an [N ii] temperature of 10 300 K derived by Kaler (1986); otherwise, we assumed a temperature of 10 000 K.
The [S iii] temperature derived for the four PNe (PN9, PN11, PN13 and PN17) is in general meaningfully different from the [O iii] temperature, and thus were used to calculate the S 2+ /H + ratio.For the other PNe, we adopted the electron temperature that was used to calculate the N + /H + ratio to derive S 2+ /H + .In the cal-culations of S + /H + , we adopted the [S ii] temperature (9200 K) for PN12; for the other targets, we adopted the temperature that was used to calculate the N + /H + ratio.Although temperatures (or the lower limits) were also derived from the [O ii] λ3727/(λ7320+λ7330) line ratio, they were much too different from the [S iii] temperatures (Table 4), given that the ionization potential of O + (35.12 eV) is close to that of S 2+ (34.83 eV); the differences between the [O ii] temperatures and those derived from the [O iii] lines are also questionable.The only exception is PN17, whose [O ii] temperature seems reasonable compared to those derived from the [O iii] and [S iii] line ratios.We adopted the electron density derived from the [S ii] λ6716/λ6731 ratio for the ionic-abundance calculations of the low-ionization species.Where available, the density yielded by [Ar iv] λ4711/λ4740 was assumed for the high-ionization species, otherwise the [S ii] density was used.For PN17, the density derived from [Cl iii] λ5517/λ5537 was used to derive Cl 2+ /H + .
Care must be taken when deriving the O + /H + ratio, although O + is not the dominant ionization stage of oxygen in PNe.We noticed that the [O ii] λ3727 (a blend of λλ3726,3729) nebular line yielded a very different O + /H + ratio from that derived from the [O ii] λλ7320,7330 auroral lines, if the same electron density was assumed.Such difference in O + /H + even reached one order of magnitude for some PNe in our sample.This might be because λ3727 and λλ7320,7330 actually come from regions with very different densi-  4 for line ratios).For PN12 and PN13, the dotted line labeled with [S ii] Te is the temperature-diagnostic curve using the [S ii] (λ6716+λ6731)/λ4072 intensity ratio.The dotted line labeled with [S ii] is density-diagnostic curve for all PNe.
ties.Besides its dependence on temperature, the [O ii] λ3727/(λ7320+λ7330) line ratio also has non-negligible dependence on the density, as can be seen in Figure 5.If these two [O ii] lines come from different nebular regions, the diagnosed temperature can be unrealistically high.
If we adopt the [S ii] density for [O ii] λ3727 and assumed a higher density (e.g., 20 000 cm −3 ; Table 4) for λ7325 (=λ7320+λ7330), the two O + /H + ratios derived can be brought to the same level (Table 5).Thus the electron densities (or the lower limit, see Table 4) yielded by the [O ii] diagnostic ratio was assumed for the λ7325 line.We then derived a weighted average from the two O + /H + ratios, with the weights proportional to the intensities of the λ3727 and λ7325 lines.The averaged O + /H + ratio was adopted and then used for the determination of elemental abundances.
Ne 2+ /H + derived from [Ne iii] λ3868 was adopted; the other [Ne iii] nebular line λ3967 is blended with H i λ3970.For the four PNe (PN9, PN11, PN13 and PN17) where both λ6312 and λ9069 of [S iii] were detected, a line intensity-weighted average of the S 2+ /H + ratios derived from the two [S iii] lines was adopted.For the other PNe in our sample, S 2+ /H + derived from λ6312 was adopted.The total intensity of the [Ar iv] λλ4711,4740 doublet was used to derive Ar 3+ /H + .The flux of [Ar iv] λ4711 was corrected for the blended He i λ4713 line.The effective recombination coefficients of the He i lines calculated by Porter et al. (2012) was used to derive the He + /H + ratios.The He 2+ /H + ratio was derived from He ii λ4686 using the hydrogenic effective recombination coefficients from Storey & Hummer (1995).We also detected C ii λ4267 (M6 3d 2 D-4f 2 F o ) in the spectra of PN11 and PN12, and the C 2+ /H + ratio was derived using this line (Table 5).The Case B effective recombination coefficients of the C ii lines were adopted from Davey et al. (2000), and an electron temperature of 10 000 K and a density of 10 4 cm −3 were assumed.
The uncertainties in ionic abundances (in the brackets in Table 5), were estimated from the measurement errors in line fluxes.Extra errors in abundances could be introduced by the electron temperatures and densities adopted in abundance determinations, although we have considered multiple ionization zones by deriving the abundances of low-and high-ionization species using different temperatures/densities.However, these errors in general have minor contribution to the total uncertainty budget, and were not included in the final uncertainties of the ionic abundances.In Paper II we have estimated that errors in the [O iii] temperature typically introduce ∼10% uncertainties in the resultant ionic abundances, while in this work such errors are probably even lower.It is evident that the ORLs of heavy elements (C ii, O ii, N ii and Ne ii) observed in PNe could be emitted by nebular regions as cold as 1000 K (e.g., Liu 2012;Fang & Liu 2013;McNabb et al. 2013).For an ORL (like C ii λ4267) excited by radiative recombination, its emissivity (i.e., effective recombination coefficient) generally decreases with the electron temperature (e.g., Osterbrock & Ferland 2006;Fang & Liu 2013).Thus C 2+ /H + derived here could be overestimated due to the temperature (10 000 K) assumed.According to the calculations of Davey et al. (2000), the effective recombination coefficient of the C ii λ4267 line decreases by a factor of 9.4 as the temperature increases from 1000 K to 10 000 K.

Elemental Abundances
The total elemental abundances (relative to hydrogen) were derived based on the ionic abundances presented in Table 5.The helium abundance is a sum of the ionic ratios, He/H = He + /H + + He 2+ /H + .For heavy elements, the total abundances were derived mostly using the ionization correction factors (ICFs) of Kingsburgh & Barlow (1994).Elemental abundances are presented in Table 6, and the ICFs used to correct for the unseen ions are presented in Table 7.
In the cases where both S + /H + and S 2+ /H + were derived, S/H = ICF(S)×(S + /H + + S 2+ /H + ) was used, where ICF(S) was adopted from Kingsburgh & Barlow (1994, Equation A36 therein).For PN16 where only S + /H + was obtainable, the empirical fitting formula of Kingsburgh & Barlow (1994, Equation A38 therein) was used to derive S 2+ /H + .If both Ar 2+ and Ar 3+ were observed, we used Ar/H = ICF(Ar)×(Ar 2+ /H + + Ar 3+ /H + ), where ICF(Ar) is from Kingsburgh & Barlow (1994, Equation A30 therein).In typical physical conditions of PNe, concentration of argon in Ar 4+ is supposed to be negligible compared to Ar 2+ and Ar 3+ .If only Ar 2+ was observed (in PN12, PN13, PN14 and PN16 in our sample), Equation A32 in Kingsburgh & Barlow (1994) was used to derive ICF(Ar).Only Cl 2+ was observed in our spectra (of PN8, PN11, PN12 and PN17), and we assumed Cl/Cl 2+ ≈S/S 2+ as in Wang & Liu (2007), according to similarity in ionization potentials.C/H was derived for two PNe in our sample assuming ICF(C) = O/O 2+ (Table 6).Delgado-Inglada et al. (2014) developed a new set of formulae for the ICFs of PNe by computing a large grid of photoionization models.These ICFs have validity application ranges defined by the ionic fractions of helium, He 2+ /(He + + He 2+ ), and oxygen, O 2+ /(O + + O 2+ ).We expect that these ICFs are adequate estimates of elemental abundances in PNe.However, not all our targets have the helium or oxygen ionic fractions located within these validity ranges.Besides, the new ICFs do not differ much from the classical methods of Kingsburgh & Barlow (1994) for most of the elements (García-Rojas et al. 2016).In particular, adopting the new ICFs of Delgado-Inglada et al. (2014) did not eliminate the "sulfur anomaly" in PNe, which is discussed in Section 4.1.
The uncertainties in the brackets following the elemental abundances in Table 6 were estimated from the errors in ionic abundances (Table 5) through propagation.The possible errors brought about by ionization corrections were not considered, although they could be significant for some heavy elements.The dominant ionization stage of helium in PNe is He + , and thus error in He/H mainly comes from He + /H + .The [O ii] and [O iii] nebular lines are among the best observed in the optical spectrum of a PN.Although determination of O + /H + is usually less accurate than O 2+ /H + due to the detection sensitivity of instruments in the optical wavelength region, concentration of oxygen in O 2+ is much higher than in O + .Besides, ICF(O) is always close to unity (Table 7).He/H and O/H thus are the most accurate among all elements analyzed in this paper.N/H and Ne/H were derived based on the ionic and elemental abundances of oxygen and are expected to be reliable.Determinations of S/H in this work were improved for some PNe by using the 2.26(±0.50)×10−5 1.53(±0.34)×10−5 8.30(±1.83)×10−6 6.09(±1.30)×10−6 1.02(±0.20)×10−5 6.89(±0.74)×10−6 1.01(±0.12)×10−5 4.12(±0.44)×10−6 5.05(±0.54)×10 4.69(±1.17)×10−6 3.57(±0.31)×10 7.32(±1.10)×10−6 1.98(±0.54)×10−6 7.89(±1.85 a The He + /H + abundance ratio derived from the He i λ5876 line is adopted.
b The N + /H + abundance ratio derived from [N ii] λ6583 is adopted.
c A weighted average value of the O + /H + ratios derived from the [O ii] λ3727 nebular and λ7325 (= λ7320+λ7330) auroral lines are adopted, with the assigned weights proportional to the intensities of these two lines.See text for details.
d The O 2+ /H + ratio derived from [O iii] λ5007 is adopted.
e The Ne 2+ /H + ratio derived from [Ne iii] λ3868 is adopted.
f For PN9, PN11, PN13 and PN17, where both the λ6312 and λ9069 lines are observed, a weighted average of the S 2+ /H + ratios derived from these two [S iii] lines is adopted, with the weights proportional to the dereddened line fluxes.
g The Ar 2+ /H + ratio derived from [Ar iii] λ7136 is adopted, because measurements of this line are much better than [Ar iii] λ7751 lying at the red end of R1000B grism.Uncertainty in carbon is also considerable since only the weak C ii λ4267 line was observed in our GTC spectrum.

Emission Features of [WR] Central Stars
We detected the C iv λ5805 (a blend of the λλ5801.33,5811.987doublet) broad emission in the spectra of PN9 and PN15 (Figure 6).Including target PN2 in Paper II, we have now observed this C iv line in three PNe in our GTC sample.In PN9, we also detected C iii λ4649 (a blend of λλ4647.42,4650.25,4651.47 of M1 2s3s 3 S-2s3p 3 P o multiplet), whose observed flux is 2.2×10 −17 erg cm −2 s −1 and line width is ∼15 Å.This C iii line is probably also blended with the faint O ii λλ4649.13,4650.84ORLs of M1 2p 2 3s 4 P-2p 2 3p 4 D o multiplet, because we found a nearby narrow emission feature which could be due to λ4661.63 of the same multiplet of O ii.Line fluxes and widths of C iv λ5805 of the three PNe are presented in Table 8.
The broad C iii and C iv lines observed in PN2, PN9 and PN15 are from their central stars, which are probably of Wolf-Rayet ([WR]) type.The intensity ratio and FWHM of C iii λ4649 and C iv λ5805 indicate that PN9 has a [WC4] central star, according to the classification of Acker & Neiner (2003, also Crowther et al. 1998).The estimated stellar temperature of PN9 seems to be slightly higher than the temperature range (∼55 000-91 000 K, Acker & Neiner 2003) covered by the [WC4]-type stars.C iii λ4649 was not observed in the other two PNe.Previous GTC spectroscopy have found [WC4]-type central stars in two outer-disk PNe (Balick et al. 2013, PN ID 174 and 2496in Merrett et al. 2006).
How these [WC] central stars formed is not well understood, although a significant fraction of Galactic PNe have been observed to harbor such type of stars.The five M31 PNe with unambiguous detection of broad [WC] features are all bright in [O iii], <0.9 mag from the PNLF bright cut-off, indicating that He-burning cores might produce visible PNe.This may help to constrain the post-AGB evolutionary models of He burners.As suggested by Balick et al. (2013), an AGB final thermal pulse, or a late thermal pulse early in post-AGB, might be the channel to these [WC] central stars.
3.6.Duplication with Recent GTC Spectroscopy PN14 in our sample was also observed at the GTC by Corradi et al. (2015, PN ID M2507), who also used the OSIRIS R1000B grism but with a slit width of 0. 8. Our observing conditions are better (seeing ∼0. 8 and clear nights).The logarithmic extinction parameters c(Hβ) derived from the two observations are also quite similar.Although the extinction laws adopted are different (Cardelli et al. 1989in this work, and Savage & Mathis 1979in Corradi et al. 2015), it has been proved that in the wavelength range covered by the OSIRIS R1000B grism, line fluxes corrected using different extinction laws differ very little.Our observed Hβ flux of PN14 is only 4.6% lower than that given by Corradi et al. (2015)

. Correlation Studies of Abundances
The abundances of α-elements (O, Ne, S, Ar, etc.) of a PN reflect the metallicity (Z) of the interstellar medium (ISM) from which the progenitor star of the PN formed.The He/H and N/O ratios are also indicative of progenitor masses: Type I PNe tend to have higher abundance ratios, while Type II PNe generally exhibit low N/O and less massive progenitors (Peimbert 1978;Maciel 1992;Kingsburgh & Barlow 1994).The α-elements of PNe also help to constrain the theory of stellar models and probe into the evolution of low-to intermediate-mass stars and consequently chemical evolution of galaxies (e.g., Kwitter & Henry 2001;Milingo et al. 2002a,b;Kwitter et al. 2003;Henry et al. 2004).Although the exact mechanism(s) still unclear, previous studies of Galactic and Magellanic Cloud PNe have established that PN morphology is a useful indicator of stellar populations (Peimbert & Torres-Peimbert 1983; Manchado et al. 2000;Shaw et al. 2001;Stanghellini et al. 2002aStanghellini et al. ,b, 2003Stanghellini et al. , 2006)).At the distance of M31, all PNe are spatially unresolved, and their central stars cannot be directly observed.The stellar population of PNe can be inferred from the elemental abundances of the nebulae.Following the discussion in Paper II, in this section we made a correlation study of the abundances of He, N, O and α-elements for our extended halo sample (see Figures 7-11).The M31 outer-disk PNe observed by Kwitter et al. (2012), Balick et al. (2013) and Corradi et al. (2015) 8 using large telescopes (Gemini-North and the GTC) were included, and henceforth mentioned as the disk sample (or the disk PNe) in this paper.Also considered in the study were previous observations of PNe and H ii regions in M31, including the bulge and disk PNe from Jacoby & Ciardullo (1999), and the nine H ii regions from Zurita & Bresolin (2012) where the temperaturediagnostic auroral lines were observed.The solar abundances from Asplund et al. (2009) and the abundances of the Orion Nebula from Esteban et al. (2004) were used as benchmarks.
Figure 7 shows the N/O abundance ratio versus He/H (left) and O/H (right) in logarithm.Our sample (the color-filled circles in Figure 7), including the PNe studied in Papers I and II, all have low N/O (<0.5) except PN16, a PN associated with M32 and whose N/O (∼1.07±0.24) is higher than the other targets, and hints at the possibility of Type I nature.Its He/H (=0.114) however, is normal compared to others.Our targets and the disk sample show no obvious trend in N/O versus He/H or O/H, and are clearly separated from the Galactic Type I objects of Milingo et al. (2010), whose N/O seems to be correlated with He/H and anti-correlated with O/H.Among the M31 disk sample, there are two outliers and one of them has N/O close to 1.0.Most of the M31 PNe have higher N/O ratios than H ii regions (including the Orion Nebula), but there are three PNe in our sample with very low N/O.Within our sample, the PNe associated with the Northern Spur, the Giant Stream, and the outer halo generally cannot be distinguished from each other in abundance ratios, although the halo target PN13 has the lowest N/O ratio.
All M31 PNe show positive correlation between neon and oxygen (Figure 8), consistent with the previous observations of PNe in the Galaxy, the Large and Small Magellanic Clouds and M31 (Henry 1989).One outlier is from Jacoby & Ciardullo (1999).This neon-oxygen posi-tive correlation was defined by the samples of H ii regions and metal-poor blue compact galaxies analyzed by Izotov & Thuan (1999), Izotov et al. (2012) and Kennicutt et al. (2003).We noticed that 12+log(Ne/H) of the Sun (7.93±0.10,Asplund et al. 2009) is slightly lower than, although still agrees within the errors with, what is expected from the neon-oxygen correlation, indicating that the current solar neon might be underestimated.This problem was investigated through a comparison study of PNe and H ii regions by Wang & Liu (2008).
Argon of our targets are generally correlated with oxygen, although the scatter is noticeable (Figure 9).Our sample also seems to have slightly lower argon than what is expected from the argon-oxygen correlation defined by the H ii regions and metal-poor blue compact galaxies, but this deficiency is more obvious in the outerdisk sample, especially in the PNe with lower oxygen [12+log(O/H)<8.45].As discussed in Paper II, sulfur abundances of M31 PNe are all lower than what is expected from the sulfur-oxygen correlation.Although the GTC spectra of four PNe (PN9, PN11, PN13 and PN17) in our sample have covered the [S iii] λλ9069,9531 lines, their 12+log(S/H) are still underabundant by ∼0.24-0.39dex.This deficiency in sulfur, known as the "sulfur anomaly", was previously noticed in the Galactic PNe (Henry et al. 2004;Milingo et al. 2010).So far the most plausible explanation seems to be the inadequacy in ICF used to correct for the sulfur ions (e.g., S 3+ and higher ionization stages) unobserved in the optical but detectable in the IR, although taking into account the IR observations of S 3+ has alleviated but could not eliminate this deficiency (Henry et al. 2012).Theoretical studies shows that sulfur is unlikely destroyed by the nucleosynthetic processes in the low-to intermediate-mass stars (Shingles & Karakas 2013).On the other hand, the M31 H ii regions of Zurita & Bresolin (2012) generally agree with the sulfur-oxygen correlation (Figure 10).
Although strictly speaking chlorine cannot be classified as an α-element because its two stable isotopes, 35 Cl and 37 Cl, are not formed through the α-processes but produced during both hydrostatic and explosive oxygen burning (Woosley & Weaver 1995), it is a secondary product during oxygen burning and created from the isotopes of sulfur and argon (Clayton 2003).Cl/H was derived for five objects in our GTC sample using the [Cl iii] λλ5517,5537 doublet (Table 6).It has also been derived for three outer-disk PNe.Together with the Galactic samples from Henry et al. (2004Henry et al. ( , 2010) ) and Milingo et al. (2010), chlorine exhibits a loose correlation with oxygen (Figure 11).The chlorine-oxygen relation among the M31 PNe alone has large scatter, probably much affected by large uncertainties, given the weakness of the [Cl iii] lines.Using the Galactic H ii regions from Esteban et al. (2015, including the Orion Nebula) as a baseline for correlation, we found that not all M31 PNe are located along this relation within the errors.

Populations of PNe
In this section, we study the stellar populations of our sample by constraining the central star parameters, which again, were estimated from the observed nebular spectra.In a PN spectrum, the intensities of nebular emission lines, such as [O iii] λ5007 and He ii λ4686, relative to the Hβ, are to some extent representative of the central star temperature (T eff ).For an optically thick PN, at a given T eff , the central star luminosity (L * ) can be determined from the nebular Hβ luminosity (e.g., Osterbrock & Ferland 2006).Based on the photoionization models of a large sample of optically thick PNe in the Large and Small Magellanic Clouds (hereafter, LMC and SMC), Dopita & Meatheringham (1991, also Meatheringham & Dopita 1991) derived empirical relationships between T eff and L * and excitation class (EC) in the form of polynomials (Equations 3.1 and 3.2 in Dopita & Meatheringham 1991).These empirical relations work in equivalent to the transformation between the observed Hertzsprung-Russell (H-R) diagram, log L(Hβ) versus EC, and the true H-R diagram, log (L * /L ) versus log T eff .The EC parameter was defined in terms of the λ5007/Hβ and λ4686/Hβ nebular line ratios.However, we were aware that these relationships were based on the models of the optically thick PNe in the Clouds, which are both metal-poor (Z=0.008 in the LMC, and 0.004 in the SMC), while previous and our current spectroscopic observations have demonstrated that the bright PNe in the outer-disk and the halo of M31 are all metal-rich (Kwitter et al. 2012;Balick et al. 2013;Fang et al. 2013Fang et al. , 2015;;Corradi et al. 2015).The relationships given by Dopita & Meatheringham (1991) could be metallicity dependent (Dopita et al. 1992), although the [O iii]/Hβ line ratio is less dependent than other lines.Thus whether they are applicable to the PNe in M31 might be questionable.
In order to assess the applicability of the relationship of Dopita & Meatheringham (1991), we derived the T eff for the M31 disk PNe studied by Kwitter et al. (2012) and Balick et al. (2013) using the relation, and compared them with the cloudy model results presented in these two papers.The differences in log T eff are mostly less than 0.06 dex.We made the same comparison of the L * of the same disk sample, and found that the differences in log (L * /L ) are mostly less than 0.05 dex.The disk sample contains the brightest PNe in M31 with m λ5007 ∼20.4-20.9.This comparison study between the two sets of T eff and L * thus confirms our anticipation that the brightest PNe in M31, those within two magnitudes from the bright cut-off of the PNLF, should not be quite evolved and probably still optically thick (A. A. Zijlstra, private communications).We noticed that the empirically derived L * of three less bright PNe (with m λ5007 = 20.72,20.88 and 20.89) are lower than their corresponding cloudy models by ∼0.2 dex.This difference might be due to a possibility that the empirical relationship of Dopita & Meatheringham (1991) underestimates the stellar luminosities of fainter PNe, which might no longer be optically thick.We made a similar comparison study  of the stellar parameters for the M31 bulge and disk PNe of Jacoby & Ciardullo (1999), which are systematically fainter (m λ5007 ∼20.73-23.16),and found that the empirically derived log (L * /L ) are lower than the photoionization models by 0.2 dex in average (although scatter in the sample of Jacoby & Ciardullo 1999 is large due to faintness of the targets).The largest difference in both log T eff and log (L * /L ) is found in the brightest PN of Balick et al. (2013, PN ID M2496, m λ5007 =20.42).
The EC of our target PNe are mostly between ∼3 and 7.6.We derived their central star temperatures and lu-minosities using the relationships of Dopita & Meatheringham (1991).Despite the applicability of these relationships assessed above, the stellar luminosity thus derived might still be underestimated for 1) bright, young (and thus probably dusty) PNe due to absorption of a large fraction of stellar flux by dust, and 2) the PNe that are optically thin to the ionizing radiation of the central stars.Since the extinction of our targets are low (Section 3.1), the first possibility could be discounted.The relatively faint PNe in M31 might not be optically thick, and their stellar luminosities derived using the empirical   Henry et al. (2004Henry et al. ( , 2010) ) and Milingo et al. (2010); the open diamonds are the Galactic H ii regions studied by Esteban et al. (2015), and the black line is a linear fit to this sample, with a slop of 1.11±0.15.The other symbols and color codes are the same as in Figure 7.
relationship might be underestimated.According to the discussion above, we revised up the log (L * /L ) of the fainter targets (m λ5007 ≥21) in our sample by 0.2 dex to compensate for the possible underestimation.The stellar temperatures and luminosities of our PNe are presented in Table 9 (the second and third columns).
The model-based log (L * /L ) versus EC relation of Dopita & Meatheringham (1991) actually is similar to the method of Zijlstra & Pottasch (1989), who derived the central star luminosities of PNe using the Hβ flux only.With the assumption that PNe are optically thick to the ionizing stellar radiation, the methodology of Zijlstra & Pottasch (1989) is analogous to the Zanstra method for determining stellar temperatures (Zanstra 1931).This assumption is justified by the very high optical depth of the H i Lyman emission lines: each recombination of hydrogen (H + + e − ) eventually results in emission of Balmer lines and the Lyα photon (Osterbrock & Ferland 2006).Following the method of Zijlstra & Pottasch (1989) and the tabulation of T eff versus L * in Pottasch (1984, p.169), we derived the stellar luminosities that are close to those derived using the empirical relationship of Dopita & Meatheringham (1991).
The log T eff versus EC relation of Dopita & Meatheringham (1991) is dependent on the spectral energy distribution (SED) of the ionizing source.Blackbodies (i.e., Planck function), as assumed in the photoionization models of Dopita & Meatheringham (1991), can result in different log T eff versus EC relations from the cases where more realistic models (like those of Rauch 2003) are used.At a given intensity ratio of two nebular lines, e.g., I(He ii λ4686)/I(Hβ), the stellar atmospheres of Rauch ( 2003) can predict higher T eff than does a blackbody.In this regard, we tried to constrain the central star temperatures and luminosities by fitting the observed line ratios (including λ4363/λ5007, He i/He ii, and [S ii]/[S iii]) to those predicted by the photoionization models calculated at a huge number of grids, the Mexican Million Models dataBase (3MdB, Morisset et al. 2015).The 3MdB models were computed with the cloudy program (Ferland et al. 2013) and include both the radiation-and the matterbounded cases, as well as stellar SEDs of both blackbody and Rauch's atmospheres.The tolerance of fitting was set to be ≤20% for all line ratios.Using the 3MdB, we managed to obtain T eff for nine objects in our sample, and they generally agree within the errors with the cor- -All masses are in unit of solar mass (M ) and ages in giga years (Gyr).a For each PN, M ini values in the first and second lines were derived using Equations 1 and 2 of the linear initial-final mass relations of Catalán et al. (2008), respectively.tms was derived based on the model grids computed by Schaller et al. (1992).
b M fin were interpolated from the post-AGB model tracks of Vassiliadis & Wood (1994).
c M fin were interpolated from the post-AGB model tracks of Miller Bertolami (2016).
responding values derived using the empirical relationship of Dopita & Meatheringham (1991).However, no optimum constraint on stellar luminosity was achieved.
The emergent spectrum of a PN can be affected by many factors, including central star parameters (T eff , L * , log g), density profile of the nebular shell, nebular abundances, nebular radius, and filling factor; fixing other parameters and varying only T eff and L * , as was usually done with cloudy for extragalactic PNe, could be unrealistic and the derived parameters problematic.In particular, determining the central star luminosity without knowing the exact covering factor (≤1) of the nebula may underestimate L * , not mentioning the case where the nebula is optically thin.
The central star locations of our PNe in the H-R diagram are shown in Figure 12, along with the new model tracks of the H-burning post-asymptotic giant branch (post-AGB) evolutionary sequences calculated by Miller Bertomali (2015Bertomali ( , 2016) ) at two metallicities, Z=0.01 and 0.02, for various initial masses. 9These two metallicities were chosen for analysis because they bracket the solar metallicity (Z ∼0.013-0.018,Grevesse & Noels 1993;Grevesse & Sauval 1998;Lodders 2003;Asplund et al. 2009) and thus are probably suitable for the metallicity environment of M31 (as discussed in Section 4.3, the O/H ratios of the total sample of M31 PNe are mostly located between the AGB models with Z=0.007 and 0.019).Also over-plotted in the H-R diagram are the 18 M31 disk PNe observed by Kwitter et al. (2012, 16 PNe) and Balick et al. (2013, two PNe).Our targets are located between the model tracks with 0.53-0.58M final masses, while the disk sample are mostly 0.56-0.657M , with a few objects extending to <0.56 M (Figure 12, bottom).The 15 disk and bulge PNe from Jacoby & Ciardullo (1999) are also presented 9 Computations of the H-burning post-AGB evolutionary track for the 1.0 M star at Z=0.01 and the isochrones presented in Figure 12 were recently updated by M. M. Miller Bertolami (K.Gesicki et al., in preparation).in Figure 12, but with large scatter in stellar luminosity.In order to make a contrast study, we also placed all samples in Figure 13 where the classical/old post-AGB evolutionary model tracks of Vassiliadis & Wood (1994) are presented.
The final core masses (M fin ) of our targets and the M31 disk sample were interpolated from the model tracks.Our targets, as constrained by the new post-AGB models, are in the range 0.536-0.583M (Table 9) with Fig. 13.-Same as Figure 14 but over-plotted with the Hburning post-AGB evolutionary models of Vassiliadis & Wood (1994, Z=0.016).The model track with 0.546 M core mass is adopted from Schönberner (1983).Red dots on the model tracks have the same meaning as in Figure 12.
an average mass of 0.562(±0.015)M , while the disk sample are mostly 0.550-0.602M with an average of 0.571(±0.016)M .Based on the old models, the core masses of our sample are 0.567-0.633M , and the disk PNe are ∼0.583-0.660M .The initial masses (M ini ) were then derived using the semi-empirical initial-final mass relation (IFMR) of white dwarfs given by Catalán et al. (2008).A comparison between the core masses derived for the two samples using the two sets of post-AGB models is presented in Figure 14 (left).
Previously, Kwitter et al. (2012) derived the M ini of their disk sample using Equation 1 in Catalán et al. (2008), which is a single linear fit to the IFMR in the 1-6 M range.In order to make a comparison study with the disk sample, we also adopted this equation for our sample.However, in the low-mass range (1-2 M ), the actual IFMR seems to be flatter than what the usual semi-empirical linear fits predict (Salaris et al. 2009;Gesicki et al. 2014).Given that our targets seem to be located between the post-AGB evolutionary tracks with the 1 and 2 M initial masses, we also derived the M ini using Equation 2 of Catalán et al. (2008), which is a linear fit for the <2.7 M range and is expected to better describe the IFMR at low masses.The M ini thus derived are also presented in Table 9 along with those derived using Equation 1 of Catalán et al. (2008).The mainsequence lifetimes (t ms ) were then derived from M ini based on the stellar model grids computed by Schaller et al. (1992, Tables 45 and 46 therein).
Our targets extend to lower core mass regions than the disk sample in the H-R diagram, and consequently older main-sequence stellar ages.Compared to the old post-AGB evolutionary models, the new models of Miller Bertolami ( 2016) have lowered the core masses and shortened the post-AGB evolutionary timescales.Us-Fig.14.-Number distributions of the core masses (M fin , left), initial masses (M ini , middle) and main-sequence ages (tms, right) of M31 PNe derived using the post-AGB evolutionary models of Miller Bertolami (2016, Z=0.02; top panels) and Vassiliadis & Wood (1994, Z=0.016; bottom panels).M ini were derived using the Equation 2 of Catalán et al. (2008).Derivation of tms is described in the text.The red-shaded histograms represent our sample, and the black-shaded histograms the disk sample.
ing the new models, we estimated the initial masses of the brightest PNe in our sample (PN9, PN10, PN11 and PN16), whose [O iii] luminosities are within 1 magnitude from the PNLF bright cut-off (Merrett et al. 2006), to be ∼1.3-1.6 M .This hints at a tantalizing possibility that the brightest PNe may also evolve from very low-mass stars.Figure 14 compares the initial masses and mainsequence stellar ages of the two samples of M31 PNe that were derived based on the two sets of post-AGB evolutionary models.It is noteworthy that according to the new post-AGB models, our sample 1) extends to core mass as low as 0.53 M (Figure 14, top-left), below the lower mass limit (0.55 M ) for the formation of PNe predicted by the old models, and 2) may evolved from young (∼2 Gyr) to intermediate age (∼6-8 Gyr) stars (while the old models place our targets in the young population, 3 Gyr).
Within our sample, we did not find obvious systematic difference in stellar mass or main-sequence age between the groups of PNe associated with different regions, although the three halo PNe (PN13, PN15 and PN17) seem to have lower stellar luminosities.According to the old stellar models, a post-AGB system with core mass lower than 0.55 M is unable to develop a visible PN because its transition time (from the beginning of post-AGB to the PN phase) is too long for the star to become hot enough ( 30 000 K) to ionize the ever-expanding nebular shell (e.g., Schönberner 1983).Updated micro-and macro-physics have been included in the new post-AGB models of Miller Bertolami (2016), which predict higher central star luminosities than the earlier models of Vassiliadis & Wood (1994, also Blöcker 1995 andSchönberner 1983) by ∼0.1-0.3 dex at given core masses, and have accelerated the post-AGB evolution by a factor of 3 to 8, enabling formation of PNe with core masses as low as ∼0.53 M .This acceleration in post-AGB evolution, which was previously proposed for the existing Blöcker (1995) model tracks by Gesicki et al. (2014) and has been well confirmed by a recent study of the central stars of 32 Galactic bulge PNe (Gesicki et al. 2017;Zijlstra et al. 2017), is more significant at lower core masses, which is reflected in that the post-AGB timescales predicted by the new models are extremely sensitive to core mass, making the low-mass PN central stars much more abundant than the more massive ones.
Most of the PNe in our sample are located before the 10 000 yr isochrone (i.e., the post-AGB evolution ages) (Figure 12).This distribution in the H-R diagram is qualitatively in line with the fact that these bright M31 PNe should not be quite evolved so that they can still be observed today.A considerable fraction of the disk PNe are located before the 5000 yr isochrone; this does not necessarily mean that they are very young PNe but might be due to overestimated stellar luminosities.In a similar sense, for those PNe located after the 10 000 yr isochrones, their stellar luminosities might be underestimated.In overall, the locations of our sample of M31 PNe in the H-R diagram are reasonable.

Comparison with AGB Model Predictions
The chemical yields of an AGB star are on dependent on its initial mass and metallicity (Z).For an AGB star with M ini ∼0.8-8 M , its surface abundances can be altered due to recurrent mixing events that bring the synthesized material to the surface (c.f., reviews Herwig 2005; Karakas & Lattanzio 2014).For stars with M ini 1.25 M , in the thermally pulsing AGB phase, instabilities in the thin He-burning shell will drive the third dredge-up, which brings the material from He-intershell and enriches the surface with carbon and s-process elements (e.g., Herwig 2005).Depending on Z, AGB stars with M ini ≥3-4 M experience the second dredge-up and hot bottom burning (HBB), which results in a significant increase in the surface nitrogen at the expense of carbon and oxygen (e.g., Karakas 2010).The nebular relative elemental abundances are thus indicative of M ini and Z. Analysis of observations against theoretical predictions provides insights to stellar astrophysics as well as assessment of the stellar evolution models.A number of theoretical efforts have been made to investigate stellar yields from AGB nucleosynthesis for various cases of M ini and Z (e.g., Herwig 2005;Karakas et al. 2009Karakas et al. , 2014;;Karakas 2010;Cristallo et al. 2011Cristallo et al. , 2015;;Lugaro et al. 2012;Ventura et al. 2013Ventura et al. , 2014;;Fishlock et al. 2014;Shingles et al. 2015;Karakas & Lugaro 2016;Pignatari et al. 2016).In this section, we make a comparison study between the elemental abundances of M31 PNe, as derived from deep spectroscopy, and AGB model predictions, aiming at constraining the initial stellar masses and investigating the dependence of stellar yields on the mass.
In Figures 15 and 16 we compare the He/H, O/H, N/O and Ne/H abundance ratios of M31 PNe to those predicted by the AGB nucleosynthesis models at different metallicities and initial stellar masses.The PN samples include our PNe (this work and Papers I and II), most of which are associated with substructures, and the disk sample observed by Kwitter et al. (2012, also Balick et al. 2013and Corradi et al. 2015).Sources of the AGB models are given in the caption of Figure 15, and cover a broad range of metallicity (Z=0.001-0.03).The He/H and O/H ratios of most PNe are located between the AGB models with Z=0.004 (SMC) and 0.014 (∼solar), although the model yields of Karakas et al. are insensitive to stellar mass at ≤3 M ; there seems to be a few objects with slightly over-solar oxygen, but still consistent with the Sun within the errors.In the AGB models of Karakas et al., N/O was assumed to be solar.Depending on the initial mass, all mixing events during the evolution of a low-to intermediate-mass star increase nitrogen by a certain amount.However, the model predicted N/O of Karakas et al. does not extend to such low levels as observed in our PN targets (Figure 15,top).It is unlikely that these AGB models over-predict N/O, given that 1) the N/O ratio only depends on the first dredgeup (because all PNe in our sample probably correspond to progenitors with M ini 2 M ) and 2) for the low-mass AGB models, predictions for the first dredge-up more or less all agree.Thus we conclude that several objects in our sample (as well as the disk sample) were probably born in the ISM with very low N/O ratios.
By contrast, N/O predicted by the AGB models of Ventura et al. (2013Ventura et al. ( , 2014) ) is very sensitive to the initial stellar mass (Figure 15,bottom).Besides, their modelpredicted N/O ratios also extend to as low as 0.02 (i.e., log(N/O)=−1.68).These differences are mainly due to the different initial compositions adopted in the two set of models, although prescriptions for convection are also different -the third dredge-up and HBB were considered at a lower stellar mass in Ventura's models (∼3 M ) than in the models of .The initial stellar masses of the M31 PNe can mostly be constrained by Ventura's models to be 3 M , consistent with the estimate of Kwitter et al. (2012, also Balick et al. 2013).The two sets of AGB models thus generally constrain the M31 PNe in metallicity and place an upper-limit on their initial masses.Ventura's AGB models also predict higher O/H at given metallicities and stellar masses, mainly due to the inclusion of convective boundary mixing (or overshooting) at different stellar evolutionary stages, especially at the thermally pulsing AGB (TP-AGB) phase, of their models that leads to the dredge up of oxygen to the surface.The broad range of Z (0.004-0.014) in M31 PNe can also be seen in the log(Ne/H) versus log(O/H) diagram (Figure 16).
Using the initial masses of the total sample of M31 PNe estimated in Section 4.2, we demonstrate the dependence of abundance ratios on M ini in Figure 17.The AGB model predictions presented in the top panels of Figures 15 and 16 are also used in Figure 17.The theoretical AGB yields from Marigo (2001) at Z=0.019 are also included.The M ini were those derived using the linear fit to the IFMR at low-masses given by Catalán et al. (2008, Equation 2 therein).For each PN, the initial masses derived based on the old post-AGB evolutionary models of Vassiliadis & Wood (1994) and the new models of Miller Bertolami (2016) were both presented.Generally, the abundance ratios of our targets and majority of the disk sample agree within the uncertainties with the theoretical AGB yields at given stellar masses.In the disk sample, there is an "outlier" with very low He/H, probably due to large uncertainty in the observations (Kwitter et al. 2012).
Interestingly, there is no obvious difference in the O/H ratios between our halo sample and the outer-disk sample, although the initial stellar masses of the disk PNe seem to be higher than those of our targets at given O/H.The O/H ratios of both samples of M31 PNe are located between the AGB models with Z=0.004 and 0.019 (Figure 17,; these two Z values correspond to [Fe/H]∼ −0.5 and ∼0.1, respectively, generally metalrich compared to the halo metallicities of M31 (Ibata et al. 2007(Ibata et al. , 2014)).Given that oxygen is the best observed element in PNe and has been used as a proxy for the metallicity of progenitor stars, from which oxygen is assumed to be inherited, this distribution of the observed O/H with respect to the AGB model predictions confirms the metal-rich nature of these M31 PNe.The insensitivity of the model-predicted N/O on stellar mass at low masses, as predicted by the AGB models of Karakas et al. (Figure 15,top), is better seen in Figure 17 (topright).This is because the first dredge-up that occurs in AGB stars with M ini ≤3-4 M does not increase nitrogen much, and oxygen is essentially unchanged.Figure 17 (top-right) also shows that significant increase in N/O occurs at M ini >3 M for the metal-poor case Z=0.001; in the metal-rich AGB stars (e.g., Z=0.014), N/O increases significantly for M ini ≥4 M (Karakas & Lugaro 2016).The M31 PNe discussed in this paper mostly have initial masses 2 M .The dispersion in Ne/H of the to-tal sample is much smaller than that in O/H, with an average ratio of 8.64(±3.35)×10−5 for our sample and 9.47(±3.34)×10−5 for the disk sample.Abundances of the PNe are mostly consistent with the AGB model predictions within the errors.
Looking at the initial masses derived from the old post-AGB evolutionary models, the N/O ratio seems to start to increase at M ini close to 2 M (Figure 17, top-right).This trend is shifted to lower masses when the new post-AGB models are used.A similar behavior in N/O has also been found in a recent study of the Galactic PNe (Henry et al. 2018).This is at variance with the current AGB model prediction that N/O increases as a consequence of HBB at M ini 3-5 M (Cristallo et al. 2011(Cristallo et al. , 2015;;Karakas et al. 2014;Karakas & Lugaro 2016;Ventura et al. 2015;Di Criscienzo et al. 2016), and tentatively implies HBB might actually occur at M ini <3 M , or even at 2 M .More detailed investigation is needed to assess this possibility.
There are three PNe with N/O>0.5 (a criterion to define the Galactic Type I PNe): PN5 and PN16 from the disk sample of Kwitter et al. (2012), and target PN16 in our sample.However, the He/H ratios of the three PNe are <0.125, at odds with the definition of a Type I PN.The initial stellar masses of the three PNe are all 2.2 M (Figure 17, top-right).Apart from the convective mixing processes (mostly the first dredge-up for low-mass stars), N/O in stars with initial masses of 1-4 M may also be affected by the non-standard mixing processes such as thermohaline mixing and stellar rotation (Karakas et al. 2009).This means that AGB stars with high N/O ratios may not necessarily have evolved from the intermediate-mass (3-8 M ) stars, where HBB is needed to enhance the surface nitrogen, but from low-mass (1-3 M ) stars that rotate reasonably rapidly on the main sequence and/or experience non-convective mixing processes during the ascent of the first giant branch.However, to what extent N/O can be affected by the thermohaline mixing, especially for the low-mass stars, is still unclear.So far very few stellar evolution models have included thermohaline mixing and rotation (e.g., Charbonnel & Lagarde 2010;Cantiello & Langer 2010).More detailed and quantitative investigation of the effects of these extra-mixing processes on the chemical yields of low-mass stars is still needed.
It is worth to mention that the solar abundances quoted in this paper are photospheric values (Asplund et al. 2009, Table 1 therein), which differ slightly from the initial abundances (or the bulk abundances; Asplund et al. 2009, Table 5 therein) due to the combined effects of gravitational settling and diffusion.When the Sun becomes a red giant, its photospheric abundances will go back to the bulk values because of convective mixing.During the post-main sequence evolution, the first and third dredge-ups change the abundances of the Sun.Stellar evolution models of low-mass (1-2 M ) stars predict that helium of the Sun can be increased by ∼0.05 dex, and N/O increased by nearly 0.3 dex (Miller Bertolami 2016).Thus the actual locations of the Sun in Figures 15,16 and 17 (also Figures 7) should be more consistent with the M31 PNe than what they appear to be now.
Finally, we emphasize that although the central star parameters of our sample were estimated using the empirical relationships, which again, were based on the photoionization modeling of the Magellanic Cloud PNe, their locations in the H-R diagram are generally reasonable, and their abundance ratio versus M ini relations are consistent with the AGB model predictions.Compared to the outer-disk sample, our halo/substructure PNe occupy the regions that correspond to relatively lower core masses in the H-R diagram, and consequently have older stellar ages.However, detailed photoionization modeling of our targets is needed to constrain the central star properties, so that a comparison study with the disk PNe can be made in a more consistent manner.This will be presented in a subsequent paper for a more extended GTC sample.The radial distribution of oxygen in M31 represented by our PNe is shown in Figure 19, where literature samples are also presented, including the M31 disk and bulge PNe from Jacoby & Ciardullo (1999) and Sanders et al. (2012), the outer-disk PNe from Kwitter et al. (2012), Balick et al. (2013) and Corradi et al. (2015), M31 H ii regions from Esteban et al. (2009, object ID K932) and Zurita & Bresolin (2012, nine objects with T e determined), and three A-F supergiants from Venn et al. (2000) and seven B-type supergiants from Trundle et al. (2002).Galactocentric distances (R gal ) have been rectified for the inclination of M31 disk (see caption of Figure 1).
In Figure 19 we mark the radii of M31's bulge, the inner/optical disk (R 25 , see caption), the halo (as well as the boundary of streams), and the extended halo.The bulge radius is adopted from Irwin et al. (2005) and well accommodates the bulge PNe studied by Jacoby & Ciardullo (1999).Almost all the disk PNe observed by Kwitter et al. (2012, also Balick et al. 2013and Corradi et al. 2015) are beyond R 25 .The oxygen abundances of M31 disk PNe seem to show a marginally negative gradient (−0.011±0.004dex kpc −1 , Kwitter et al. 2012) within 40 kpc from the centre of M31, and tend to flatten out to 100 kpc.In our halo sample, members of which are mostly associated with the substructures of M31, we also found a similar trend of flattening in oxygen.In Paper II, based on a limited sample, we drew a conclusion that the PNe in the Northern Spur and those associated with the Giant Stream have homogeneous oxygen abundances (see Figure 13 in Paper II). Including the ten new targets (PN8-PN17), our extended sample displays generally consistent oxygen abundances.
A very prominent feature in Figure 19 is the solar oxygen of the outermost nebula PN13.Its apparent galactocentric radius is 42 kpc (Table 1); if we assume that PN13 is in the extended disk of M31 (Ibata et al. 2005), its rectified radius will be ∼180 kpc, which could be set as an upper-limit distance of this PN.The 12+log(O/H) value of PN15 is 0.13 dex above the Sun (8.69, Asplund et al. 2009), a difference that is close to the typical uncertainty in oxygen of our sample.However, the exact galactocentric radii of PN13 and PN15 are unknown because of the dubious disk membership according to their kinematics (Figure 2).
Another halo object PN17 has an oxygen abundance ∼0.15 dex below the Sun.A galactocentric radius of 109 kpc is estimated for PN17, if we assume that it is in the outer disk.However, considering that its radial velocity deviates significantly from the extended disk (Figure 2, bottom), PN17 is probably a halo PN, or even associated with its substructure.PN17 is located near the NE Shelf, an overdensity of metal-rich RGB, whose stellar populations were found to be similar to those of the Giant Stream (Ferguson et al. 2005).Numerical simulations have suggested that the NE Shelf might be debris from the continuation of the Giant Stream (Ibata et al. 2004;Fardal et al. 2008;Mori & Rich 2008).Given that the oxygen abundance of PN17 is very close to that of PN7, which is well associated with the Giant Stream (Fang et al. 2015), we thus suggest that PN17 might be associated with the NE Shelf.
PN14 is located in the southeast outer halo of M31, and has a rectified galactocentric distance of 108 kpc if we assume that it is in the extended disk.Its oxygen abundance is ∼0.4 dex below the Sun.According to its radial velocity and location, we suggest that PN14 might be associated with the Giant Stream (while in Merrett et al. 2006 this PN was not identified to be related to any substructure).Although compared to PN7, PN14 is not located right on the stellar orbit proposed by Merrett et al. (2003,   PN.The galactocentric distances of PN14 and PN17 are both uncertain. Although PN8, PN9 and PN10 have been identified by Merrett et al. (2006) as in the Northern Spur (Table 1), their oxygen abundances are different: the former two PNe are both close to the Sun, while the latter is 0.23 dex below the solar value (Figure 19).These three PNe, together with the other six in the Norther Spur, form a sample in this substructure with 12+log(O/H)∼ −0.26-0.Being 0.13 dex above the Sun, the 12+log(O/H) value of PN16 is among the highest in our sample.The galactocentric distance of PN16 (22.6 kpc) was estimated using the distances to M31 (785 kpc) and M32 (763 kpc, Karachentsev et al. 2004) and its angular distance (∼0.• 4) to the centre of M31.It is interesting to note that the oxygen abundance of PN16 seems to well fit the trend of the weakly negative gradient represented by the disk PNe of Kwitter et al. (2012).
Together with the M31 outer-disk PNe, our halo targets are metal-rich ([O/H]∼ −0.4 to 0) compared to the mean metallicities of the dominant stellar populations at similar galactocentric radii in the outer halo ([Fe/H] −1.5 to ∼ −0.6; e.g., Brown et al. 2006aBrown et al. , 2008;;Chapman et al. 2006Chapman et al. , 2008;;Koch et al. 2008;Richardson et al. 2009).In contrast with the high degree of inhomogeneity in the metallicity of M31's extended halo (from the most metal-poor population in the outer regions with −2.5 <[Fe/H]< −1.7, to the relatively metal-rich population in the inner halo where −0.6 <[Fe/H] 0; Ibata et al. 2014), the averaged values of [O/H] of our halo PNe in different kinematic groups (i.e., different substructures) are quite similar, ∼ −0.2 to 0, with modest dispersion, and are consistently higher than the average [Fe/H] of stars in the respective substructures.The uniformly high oxygen abundances of our PNe indicate that they formed from metal-rich ISM, and probably belong to the population that is distinct from the underlying halo populations of M31.
All PNe selected for our spectroscopic observations are among the brightest in M31, within 2 magnitudes from the bright cut-off of the [O iii] PNLF, which allows a reliable spectral analysis and accurate abundance determinations (Section 3).Whether the metallicity of bright PNe depends on the [O iii] luminosity still needs careful studies, although this possible relation has previously been investigated for the PNe in LMC (Richer 1993), M31 (Jacoby & Ciardullo 1999), M32 (Richer & McCall 2008), M33 and the Milky Way (e.g., Magrini et al. 2004).In these galaxies, oxygen abundances of the bright PNe are found to be independent of the [O iii] absolute magnitude (M λ5007 ), and the brightest PNe are representative of the whole PN population.For the bright PNe (M λ5007 ≤ −3.7) in M31's outer disk, oxygen abundance has no dependence on M λ5007 , although a slight tendency of decreasing oxygen seems to exist in a few objects on the fainter tail (Corradi et al. 2015, Figure 4 therein).A very similar trend was found in our halo/substructure sample (Figure 20).Here M λ5007 were derived from the [O iii] apparent magnitudes (m λ5007 , adopted from Merrett et al. 2006), which were corrected for the foreground (and also internal) extinction using c(Hβ).
Moreover, as predicted by the new post-AGB evolutionary models, low-mass central stars of PNe evolve much faster than previously thought (Miller Bertolami 2016;Gesicki et al. 2017), thus increasing the possibility that lower mass (and thus relatively older) stars may also form bright PNe that are still visible.Our population analysis (Section 4.2; Table 9) also indicates that bright PNe might evolve from low-mass stars, which is different from the traditional view that bright PNe evolve from rare, high-mass progenitors (e.g., Schönberner et al. 2007).Hence, we conclude that our sample well represents the population of PNe in the halo of M31, regardless of their spatial locations and kinematics.As one of the most efficient coolants in PNe, the [O iii] λ5007 luminosity is highly dependent on the central star effective temperature (Dopita et al. 1992;Schönberner et al. 2007), but is also regulated by the nebular metallicity (in particular, the fraction of oxygen in O 2+ ); thus slight dependency of the [O iii] λ5007 luminosity on oxygen abundance may exist.
The discussion above is based on the assumption that oxygen is neither created nor destroyed during the evolution of PN progenitors (in our case, low-mass stars; Section 4.2), although third dredge-up and HBB might enhance the oxygen of some AGB stars (e.g., Karakas & Lattanzio 2014;Delgado-Inglada et al. 2015).In order to compare with M31's halo metallicities indicated by [Fe/H], we also assumed that [Fe/O]≈0 (i.e., the solar case) for the bright PNe in M31.

Possible origin of Luminous PNe
in the Halo of M31 Our previous studies of the M31 halo PNe associated with substructures were based on two a priori assumptions: 1) these PNe represent the stellar populations of the substructures where they are located (i.e., they formed in situ); and 2) the PNe associated with substructures have different origins -and thus probably belong to different populations -from those in the disk of M31.However, so far we did not find any discernible difference between our halo sample and the outer-disk PNe in the radial distribution of oxygen at R gal 110 kpc (Figure 19).
Based on the old post-AGB evolutionary models, as were adopted by Kwitter et al. (2012), the stellar ages of our halo sample and the outer-disk sample are both constrained to be 2-3 Gyr.These M31 PNe thus are the young population, with their stellar ages in line with the episode of star formation that occurred across the whole disk system of M31 2-4 Gyr ago (e.g., Richardson et al. 2008;Bernard et al. 2012Bernard et al. , 2015a,b;,b;Williams et al. 2015).The onset of this recent global starburst may correspond to an encounter with M33, whose stellar disk also experienced enhanced star formation ∼2 Gyr ago (Williams et al. 2009;Bernard et al. 2012).This M31-M33 interaction also explains the stellar streams seen in the M31 halo (McConnachie et al. 2009), and has been invoked to account for the luminous, oxygen-rich PNe in M31's outer disk (Balick et al. 2013;Corradi et al. 2015).
However, using the well assessed new post-AGB evolutionary models, we confined the main-sequence ages of our halo sample to be mostly ∼2-5 Gyr, with the oldest being ∼6-8 Gyr; while the outer-disk sample are mostly 1-4 Gyr.We thus conjecture that our targets probably formed prior to the encounter with M33.Obviously, our sample represents the population that is different from the underlying, smooth, extended (and mostly metalpoor) halo component of M31 (Ibata et al. 2007(Ibata et al. , 2014)), which was formed through repeated accretion of smaller galaxies in the distant past.These bright PNe seem to resemble the younger, metal-rich population in the outer stream of M31, as revealed by HST pencil-beam pointings on the Giant Stream (Brown et al. 2006a;Bernard et al. 2015a).The metallicity of the stream fields enriched continuously from [Fe/H]∼ −1.5, to at least solar level about 5 Gyr ago (Bernard et al. 2015a).This timeline of metal enrichment is generally consistent with the stellar ages of our metal-rich sample.N -body simulations suggested that the Giant Stream and other stream-like features in the halo are debris of a massive ( 10 9 -10 10 M ) progenitor that was recently disrupted during the course of merger (e.g., Ibata et al. 2004;Geehan et al. 2006;Font et al. 2006;Fardal et al. 2006Fardal et al. , 2007Fardal et al. , 2008Fardal et al. , 2013;;Mori & Rich 2008;Sadoun et al. 2014).The extended star-formation history and a broad range of metallicity (−1.5 [Fe/H] 0.2) discovered in the stream fields can be explained by a disk galaxy progenitor (Brown et al. 2006a,b;Bernard et al. 2015a).If the stellar streams in M31's halo indeed have a common origin, our sample of halo PNe then probably formed through extended star formation in this possibly massive, disk-like progenitor.Moreover, some simulations predict that the remnant of the disrupted satellite resides in the NE Shelf (e.g., Fardal et al. 2008Fardal et al. , 2013;;Sadoun et al. 2014); PN17 in our sample is located in this region and might be associated with this substructure (see Section 4.4).
On the other hand, despite the systematic discordance in the estimated stellar ages (and also in kinematics) between the two samples of M31 PNe, their consistently high oxygen abundances (see Figure 19) signifies something maybe in common.Recently, it was suggested that the PNe associated with the outer stellar streams might have formed from the same metal-rich ISM as did the outer-disk sample, but acquired the kinematics of the streams during a subsequent encounter with M33 (Balick et al. 2017).This postulation hints at a possibility that the halo PNe on the streams might have their origins in the M31 disk, where the metal-rich ISM mostly resides.The close interaction between M31 and M33, combined with the recent impact of the Giant Stream's progenitor, could heat up the main disk and redistribute the disk material into the substructures we see today (Bernard et al. 2015a).
The interpretations presented above are highly speculative, given that the M31 halo is extremely complex and the samples of PNe with high-quality spectroscopic observations are still very limited.In addition, Magrini et al. (2016) suggested that radial migration of the innerdisk PNe in M31 could be important and explain the flattening of oxygen gradient (compared to the H ii regions).This migration mechanism unlikely occurred to our halo PNe because to fling a star inside the disk into halo regions far beyond R 25 requires many orbital scatterings due to gravitational anomalies, such as dense spiral arms (e.g, Binney & Tremaine 2008).
A recent proper motion analysis and cosmological simulations of the M31-M33 system suggests it is unlikely that M33 made a recent (<3 Gyr), close (<100 kpc) passage about M31 (Patel et al. 2017); this is inconsistent with the scenario proposed by McConnachie et al. (2009, also Putman et al. 2009).If true, the interactions between M31 and the Giant Stream's progenitor then seem more plausible to explain the kinematics of the metal-rich PNe in the M31 halo.
M32 was speculated to be responsible for the Giant Stream (Ibata et al. 2001a;Ferguson et al. 2002;Choi et al. 2002;Merrett et al. 2003), but kinematical studies ruled out this possibility (Ibata et al. 2004).The stellar orbit proposed by (Merrett et al. 2003; see Figure 2) that links the Giant Stream to the Northern Spur is rather generic yet highly schematic.The light-of-sight distance to M32, although previously derived (e.g., Jensen et al. 2003;Karachentsev et al. 2004), is still uncertain.The N/O ratio of PN16 is higher than our halo PNe, also casting doubts on the previous hypothesis that M32 was the progenitor of substructures.

Comments on Individual Objects
Several PNe in our GTC sample are interesting in terms of abundances, spatial locations, and kinematics, and worth extra attention.Although most of these PNe have already been discussed in previous sections, their main characteristics are briefly emphasized here.
PN7 -The mostly distant PN so far studied in the extended halo of M31.Well located in the southeast extension of the Giant Stream and with an oxygen abundance close to the Sun, PN7 is an archetypal object that represents the group of metal-rich nebulae associated with the outer-halo streams at large galactocentric radii.
PN13 -An outer-halo PN spatially located on the Giant Stream.If we assume that PN13 is in the extended disk, its rectified galactocentric distance will be ∼180 kpc, making it the most distant solar-metallicity PN in M31 so far observed.However, its disk membership is highly questionable due to kinematics.
PN14 -It might be associated with the Giant Stream.Its oxygen is ∼0.4 dex below the Sun, but is close to those of PN4 and PN12 on the stream.
PN15 -Its oxygen is ∼0.13 dex above the Sun, mak-ing it the most metal-rich PN so far observed in the outer halo of M31.Same as PN13, its actual distance to the centre of M31 is unclear.PN16 -The only M32 PN analyzed in this work.Its O/H and N/O ratios are both higher than the other PNe in our sample.PN16 also has the highest oxygen abundance among all PNe so far spectroscopically studied in M32 (Richer et al. 1999).The estimated stellar age (t ms ∼2 Gyr) of PN16 is consistent with the younger stellar population discovered in M32 (2-5 Gyr, Monachesi et al. 2012).This bright PN (m λ5007 =20.78) probably well represents the young population of M32.Given that currently reliable abundance measurements of the M32 PNe are extremely sparse, our observations provide valuable nebular abundances of this dwarf elliptical galaxy.
PN17 -This halo PN might be associated with the NE Shelf, as judged from its O/H, spatial position, and kinematics (see the discussion in Section 4.4).If it is true, PN17 will be the first PN discovered and studied in this halo substructure.

SUMMARY, CONCLUSION
AND FUTURE WORK With the aim of studying the properties and possible origins of stellar substructures in the halo of M31, we carry out deep spectroscopy of PNe using the 10.4 m GTC.Following our previous effort (Fang et al. 2015), where seven PNe associated with the Northern Spur and the Giant Stream were targeted, we obtained highquality GTC long-slit spectra of ten PNe that reside at different regions and cover a vast area of the M31 system.These new targets are associated with the two known substructures, the eastern and southern regions of M31's halo, and dwarf satellite M32.The OSIRIS spectrograph secures a wavelength coverage of ∼3630-7850 Å and detection of a number of plasma-diagnostic emission lines, including the temperature-sensitive [O iii] λ4363 and [S iii] λ6312 lines and the density-diagnostic [S ii] λλ6716,6731 doublet.We also observed the [S iii] λλ9069,9531 nebular lines in four PNe, whose GTC spectrum extends beyond 1 µm.Ionic and elemental abundances (relative to hydrogen) of helium, oxygen, nitrogen, neon, sulfur, argon and chlorine were derived using different temperatures and densities according to ionization stages.
The N/O ratios of our halo sample are mostly <0.4, and He/H<0.126,indicating that they might be Type II, i.e., of relatively low-mass progenitors.These abundance are generally consistent with the outer-disk sample recently observed with the 8-10 m telescopes (Kwitter et al. 2012;Balick et al. 2013;Corradi et al. 2015).In both samples, Ne/H is well correlated with O/H; in some objects, argon is under-abundant with respect to the Ar/H versus O/H correlation.The "sulfur anomaly", originally found in Galactic PNe, also exists in the M31 PNe, even for the objects with detection of the [S iii] λλ9069,9531 lines.Although with large scatter, Cl/H of our targets are generally in line with the Cl/H versus O/H correlation as defined by the Galactic H ii regions with best measurements of chlorine.In an M32 PN, we found a relatively high N/O ratio (∼1.07±0.24).
The central star temperatures and luminosities of our targets were derived using the empirical method.In the H-R diagram, our sample occupies the regions that cor-respond to relatively lower core masses than does the outer-disk sample.As constrained using the new post-AGB evolutionary models, which have recently been assessed through studies of Galactic PNe, the core masses of our sample of halo PNe are 0.53-0.58M , and those of the outer-disk sample are 0.55-0.64M ; the mainsequence ages of our targets are then mostly ∼2-5 Gyr, with the oldest being ∼6-8 Gyr; while the outer-disk PNe are mostly ∼1-4 Gyr.If the new post-AGB evolutionary models are adopted in our population analysis, the PNe so far observed in M31 probably all evolved from low-mass ( 2.2 M ) progenitors, which formed from the metal-rich ([O/H] −0.4) ISM.In the H-R diagram, our GTC targets are mostly located before the theoretical isochrone of 10 000 yr since the beginning of post-AGB evolution; given that they are the brightest PNe in M31, their locations in the diagram are generally consistent with the high probability that these nebulae are not quite evolved in the PN phase and still stay around their peak [O iii] luminosities.Our estimated central star parameters thus are, although still preliminary, generally reasonable and suggest that the brightest PNe in M31 are probably optically thick.
The He/H, O/H, N/O and Ne/H ratios of both samples of M31 PNe were compared with the AGB nucleosynthesis models, and general consistency between the observations and theoretical predictions of AGB yields was found.From the model-predicted abundance ratios, we constrained the upper-limits of the initial stellar masses of both samples of M31 PNe to be <3 M , in line with our mass estimate based on the new post-AGB evolutionary models.We also studied the dependence of abundance ratios on initial mass, and found in overall excellent agreement.In particular, as a stellar-mass indicator, the observed N/O ratios mostly agree, within the errors, with the AGB models of Karakas et al. at low masses (1-3 M ), except for several outliers.Although still limited by the sample size, N/O of M31 PNe seems to start to increase at 2 M , a trend similar to what has been recently found in Galactic PNe, indicating that HHB might actually occur in very low-mass stars.According to the AGB models, O/H of the combined sample of M31 PNe span a broad range that encompasses the metallicities of the SMC (Z=0.004) and the Sun (Z 0.02), irrespective of the initial masses.The O/H ratios of our halo sample are similar to those of the outer-disk PNe.From the oxygen abundance, spatial location and kinematics, we suggest that PN17 in the eastern halo might belong to the NE Shelf.
Our extended sample of the halo/substructure PNe exhibit uniformly high oxygen abundances with modest scatter.We found nearly-solar oxygen in several PNe that are located in the outer halo.The most interesting target with solar oxygen is on the southeast extension of the Giant Stream, with a sky-projected galactocentric distance of ∼50 kpc.In one of the outermost PNe, we even found slightly over-solar oxygen ([O/H]≈0.13);there is also a halo target in our GTC sample with slightly sub-solar oxygen, [O/H]∼ −0.4.Our targets probably belong to the population that is different from the underlying, smooth halo component of M31, but more like the metal-rich populations in the streams.The estimated stellar ages of our sample (mostly ∼2-5 Gyr) are consistent with the metal en-richment history recently unveiled in the stellar fields of the Giant Stream.If the substructures where our targets are associated have a common origin (i.e., they are tidal debris from a possibly massive, disk-like satellite of M31), our deep spectroscopy of nebulae confirms the extended star-formation history in this satellite that was previously unveiled through HST photometric studies.Alternatively, our observations cannot exclude the possibility that our targets might originally formed in the M31 disk, but were scattered to the halo regions and gained their current stream kinematics as a result of interactions with M31's satellite, which can heat up and perturb the stellar disk and redistribute the disk material.Either of the above two interpretations supports the current astrophysical picture that M31's halo evolved through complex galactic mergers and interactions.
Through unprecedentedly deep spectroscopy of a sample of PNe mostly associated with the substructures in the halo of M31, we have obtained intriguing results which, together with the recent spectroscopic observations of the outer-disk sample by other research groups at the 8-10 m class telescopes, may have profound effects in our understanding of the M31 system, especially its highly structured extended halo, which so far has been rarely investigated through quantitative spectroscopy of the ISM.Our findings are qualitatively consistent with the complexity of M31's giant halo.However, one still needs to realize that at this stage no definite conclusions can be drawn and all interpretations are highly speculative, given the limited PN samples and the complex evolutionary history of M31.
The number of PNe discovered and identified in the outskirts of M31 has been increased in recently years.To date our observations mainly focus on the PNe in the Northern Spur and the Giant Stream substructures that cover the southeast, the eastern and the northern regions of the halo.In the follow-up observations at the 10 m GTC, we target the western halo objects and extend further north and south, >2 • from the centre of M31.Through these combined efforts, we aim to construct a statistically significant and spatially unbiased sample for detailed analysis (including photoionization modeling), to make a census study of M31's extended halo, and eventually to obtain a grand picture of its evolution.

Fig. 2 .
Fig. 2.-Spatial and kinematical distribution of PNe in M31 (description of the X and Y coordinates is given the text).Source of the PN samples are the same as in Figure 1.The three Northern Spur PNe studied in Paper I indicated by red asterisks.The seven PNe studied in Paper II and the ten PNe in this work are highlighted (see the legend).The red crosses "×" are the outer-disk PNe studied byKwitter et al. (2012, Kwitter12),Balick et al. (2013, Balick13), andCorradi et al. (2015, Corradi15).In the upper-left panel: over-plotted is the stellar orbit proposed byMerrett et al. (2003, thick  green curve; reproduced with kind permission of the authors).In the two side panels (bottom and right): a projection of the orbit in line-of-sight velocity with respect to M31, v los , versus distance along the major and minor axes of M31 is superimposed on the PNe data.Velocities of PNe have been corrected for the systemic velocity −306 km s −1 of M31(Corbelli et al. 2010).The red dashed curve in the bottom panel is the H i rotation curve fromCarignan et al. (2006).

Fig
Fig. 3.-Negative grey-scale GTC OSIRIS g-band acquisition image of PN12 (marked with the black crosshair) taken with an exposure of 5 s.
Fig.4.-GTCOSIRIS spectrum of PN12 (top) and PN17 (bottom).The spectrum of PN12 was obtained using the R1000B grism, while the spectrum of PN17 is a combination of the R1000B (≤7500 Å) and R1000R (>7500 Å) spectra.Vertical ranges of both panels are set to accommodate the intensity of Hβ.All important emission lines are labeled.Extinction has not been corrected for.For PN12, the weak features between [O ii] λλ7320,7330 and [Ar iii] λ7751 are the second-order effect.In the spectrum of PN17, the strong emission features redward of [S iii] λ9531 are also due to the second-order contamination.

Fig. 5 .
Fig. 5.-Plasma diagnostic diagrams.Different line types represent the temperature or density diagnostics using different line ratios (see Table4for line ratios).For PN12 and PN13, the dotted line labeled with [S ii] Te is the temperature-diagnostic curve using the [S ii] (λ6716+λ6731)/λ4072 intensity ratio.The dotted line labeled with [S ii] is density-diagnostic curve for all PNe.

Fig. 6 .
Fig. 6.-Spectrum of PN9 and PN15 showing the broad C iv λ5805 emission line (middle and right panels) of the central stars.The C iii λ4649 line (left panel) might also be due to the central star emission and is fitted by a Gaussian profile with FWHM ∼15 Å.

Fig. 7 .
Fig. 7.-N/O versus He/H (left) and O/H (right), both displayed in logarithm.Different symbols represent different samples (see the legend).Explanation of different data sets are given in the text.The color-filled (blue, red, green and pink) circles are our GTC sample, including those observed in Paper II; color code is the same as in Figure 1 (see also the top-left legend).Red open circles are the Northern Spur PNe studied in Paper I. The M31 disk PNe observed by Kwitter et al. (2012, Kwitter12), Balick et al. (2013, Balick13) and Corradi et al. (2015, Corradi15) are represented by black filled circles.Typical error bars of the two samples are indicated at the corner.Symbols in Figures 8-11 have the same meaning.

Fig. 8 .
Fig. 8.-Ne/H versus O/H (left) and Ne/O versus O/H (right), both displayed in logarithm.The small black dots are the extragalactic H ii regions and metal-poor blue compact galaxies (see references in the text), and the solid black line in the left panel is a least-squares linear fit to these data.These symbols in Figures 9 and 10 have the same meaning.Color codes of the other symbols are the same as in Figure 7.

Fig. 9 .
Fig. 9.-Same as Figure 8 but for Ar/H versus O/H (left) and the Ar/O ratio versus O/H (right).

Fig. 10
Fig. 10.-Same as Figure 8 but for S/H versus O/H (left) and the S/O ratio versus O/H (right).

Fig. 11 .
Fig. 11.-The Cl/H versus O/H in logarithm scale.The small open triangles are the Galactic PN samples fromHenry et al. (2004Henry et al. ( , 2010) ) andMilingo et al. (2010); the open diamonds are the Galactic H ii regions studied byEsteban et al. (2015), and the black line is a linear fit to this sample, with a slop of 1.11±0.15.The other symbols and color codes are the same as in Figure7.

Fig. 12 .
Fig. 12.-Central star positions of M31 PNe in the log (L * /L ) vs. log T eff diagram.The black asterisks are the disk PNe observed by Kwitter et al. (2012) and Balick et al. (2013); our targets are represented by the colored asterisks (color codes are the same as in previous figures).The black open circles are the M31 bulge and disk PNe studied by Jacoby & Ciardullo (1999).Over-plotted in the diagram are the model tracks of the H-burning post-AGB sequences calculated by Miller Bertolami (2016) at Z=0.01 (top) and 0.02 (bottom); different line types represent different initial and final masses.Red curves are isochrones for the evolutionary ages (τ = 0, 5000, 10 000, 15 000, and 20 000 yr) since the zero point of post-AGB defined at log T eff =3.85.

Fig. 15 .
Fig. 15.-The N/O versus the He/H (left) and O/H (right) abundance ratios (in logarithm); only our targets and the M31 disk sample are presented, along with the abundance ratios of the sun (Asplund et al. 2009) and the Orion Nebula (Esteban et al. 2004).Symbols and color codes are the same as in Figure 7.In the top panels: AGB model predictions from Karakas (2010, Z=0.004), Fishlock et al. (2014, Z=0.001) and Karakas & Lugaro (2016, Z=0.007, 0.014 and 0.03) for the surface abundances at different metallicities are over-plotted for purpose of comparison.In the bottom panels: AGB model predictions from Ventura et al. (2013, Z=0.001) and Ventura et al. (2014, Z=0.004) are over-plotted.Different symbols represent different metallicities; symbols of the same metallicity are linked by dotted lines to aid visualization.Initial mass (in M ) of the progenitor star is labeled for each model.The vertical scales of the top and bottom panels are different to accommodate the AGB model grids.

Fig. 17 .
Fig. 17.-Abundance ratios of M31 PNe versus initial mass.Sample includes our targets and the disk PNe from Kwitter et al. (2012) and Balick et al. (2013).For each PN, the initial masses derived from the post-AGB evolutionary models of Vassiliadis & Wood (1994, filled circle) and Miller Bertolami (2016, open circle) are both presented and connected by a solid line.Color codes of data points are the same as in Figures 15 and 16.Representative abundance errors of the two samples are indicated.The AGB yields predicted at different metallicities, as represented by different line types (see the legend), are over-plotted.The AGB models are from Karakas (2010, Z=0.004), Fishlock et al. (2014, Z=0.001), Karakas & Lugaro (2016, Z=0.007, 0.014 and 0.03), and Marigo (2001, Z=0.019).The horizontal red-dotted lines mark the solar values (Asplund et al. 2009).

Fig. 18 .
Fig. 18.-Histograms of M31 PNe in the O/H (left), N/O (middle) and He/H (right) ratios in logarithm.The red-shaded histograms represent our sample, and the black-shaded histograms the disk PNe.The vertical dotted lines mark the original definition of Galactic Type I PNe by Peimbert & Torres-Peimbert (1983): N/O>0.5 or He/H>0.125. 10

4. 4 .
Radial Oxygen Abundances A general comparison of our PN sample and the M31 disk PNe in He/H, O/H and N/O (in logarithm) is presented in Figure 18.Including the three Northern Spur PNe studied in Paper I, our sample of 20 PNe has a range of 8.30-8.82 in 12+log(O/H) with an average of 8.55±0.15,close to that of the disk sample (27 PNe).Our sample has slightly larger scatter in N/O (Figure 18, middle), with an average of 0.28±0.21.

Fig. 19
Fig. 19.-Radial distribution of oxygen in M31.Galactocentric distances (R gal ) have been corrected for the inclination of M31 disk.Our GTC sample are the color-filled circles, color-coded in the same manner as in previous figures.Note that the radial distance of PN13 (∼180 kpc) is estimated assuming that it is on the extended disk, and is only an upper limit.Other data sets (see the legend) are explained in the text.The solid black line is a linear fit to the outer-disk PNe of Kwitter et al. (2012, 18 R gal 43 kpc).Typical error bars of different samples are given in the bottom-right corner.The horizontal red-dotted line marks the solar value (8.69, Asplund et al. 2009).In order for convenience in abundance comparison, the distance between 115 and 170 kpc is not properly scaled.The horizontal black-dashed and dotted lines represent the mean metallicity and dispersion of halo stars between 10 and 60 kpc (estimated from the measurements of Richardson et al. 2009).The colored bars in the figure bottom mark the radii of M31's bulge (pink; ∼3.4 kpc, Irwin et al. 2005), the optical disk (yellow; R 25 ≈22 kpc), and the outer halo as well as the farthest outreach of the Giant Stream (green; ∼8 • , corresponding to 110 kpc in projection); further out is more extended halo (light blue).
see also Figure 2), considering the large spatial extension (along the direction orthogonal to the stream) of the Giant Stream (McConnachie et al. 2003), this association is possible.As reported in Section 3.6, PN14 has been observed by Corradi et al. (2015, PN ID M2507), who used the same instrument.The difference in 12+log(O/H) between the two observations is well within the measurement uncertainty; the galactocentric distance of PN14 given by Corradi et al. (2015, 106 kpc) is slightly smaller than ours, which is due to a slightly different distance to M31 adopted.If we assume PN14 belongs to the Giant Stream, by applying its three-dimensional structure (McConnachie et al. 2003), we estimate a distance of roughly 50 kpc for this

Fig. 20 .
Fig. 20.-Oxygen abundances of M31 PNe versus the absolute magnitudes at [O iii] λ5007.Our halo/substructure PNe are red symbols, and the outer-disk PNe are black.For each PN, the [O iii] magnitudes with and without extinction correction (filled and open circles, respectively) are both presented and connected by a dash line.Solar oxygen is marked by a horizontal dotted line.

TABLE 5 (
Continued) For each PN, abundances both in linear form and logarithm, log(X/H) + 12, are presented.
iii] λ9069 nebular line and adopting the [S iii] temperatures.Errors in S/H and Ar/H introduced by the ICFs are non-negligible, but difficult to quantify.Thus they were not considered in the error estimation.Uncertainty in Cl/H is large because only Cl 2+ was observed.

TABLE 9 Estimated
Properties of PN Central Stars and the Progenitors a