The Timescale Correlation Method: Distances to Planetary Nebulae with Halos Arsen R

THE TIMESCALE CORRELATION METHOD: DISTANCES TO PLANETARY NEBULAE WITH HALOS ARSEN R. HAJIAN,1 ADAM FRANK,2 BRUCE BALICK,3 AND YERVANT TERZIAN4 Received 1996 July 15; accepted 1996 September 16

ABSTRACT We present deep narrow-band CCD images of six PNe obtained with the Palomar 5 m telescope in the light of [N II] j6584 ] Ha and [O III] j5007. Several previously undetected structures are found, including faint multiple envelope structures surrounding three of the targets. The inclusion of these sources with data for other multiple envelope PNe published in the literature permit a statistical evaluation of the relationship between PN ““ shells ÏÏ and the thermal pulses of the PN nucleus. In this paper, we investigate the dynamical timescale between successive envelopes,q , and the nuclear interpulse timescale,q . Our results show that the deviations from the relationq dyn\ q can be explained by the ip dyn ip uncertainty in the Shklovsky distance to individual PNe, which ispd /d D 2. By imposing the constraint thatq \ q , we Ðnd that it is possible to derive a PN distancesh indicator,sh which we name the timescale correlationdyn ip distance,d . The derivation ford is independent of ad hoc and often unsup- ported assumptions inherent to theTC Shklovsky method. TC Subject headings: ISM: structure È planetary nebulae: general

1. INTRODUCTION Arendt1987; Jewitt, Danielson & Kupperman 1987; Schwarz,Corradi, & Melnick 1992) show that many bright As early as 1918(Curtis 1918) it was known that some cores are surrounded by very faint, limb-brightened halos. planetary nebulae (PNe) possess multiple envelopes. Today, Such edge-brightened structures are currently interpreted as it seems that this phenomenon is prevalent in PNe (Terzian evidence supporting the notion that PNe shells, evolve 1983; Chu 1989). Some nebulae, like NGC 6543 and NGC hydrodynamically and are shaped by interactions with 6826, exhibit giant outer halos and a few others like NGC winds and the ambient interstellar medium (ISM) (Kwok 2022 and NGC 7662 show triple shells. To avoid confusion 1983; Balick 1987; Frank& Mellema 1994; Frank et al. regarding the various morphological components of PNe, 1990). Competing ballistic models of PN expansion cannot we adopt the terminology ofBalick et al. (1992) in sub- produce such structures without speciÐc and ad hoc dividing ““ envelopes ÏÏ into three groups. assumptions concerning the time dependence of the mass- 1. Cores, which consist of a thin, compact(r [ 0.03 pc), loss rate from the central star(Frank et al. 1990). Further- elliptical ring centered on and surrounding the nucleus. more,Balick (1987) and Stanghellini & Pasquali (1995) note Cores are almost always the most observable portion of the that the multiple envelopes of elliptical PNe preserve the PN system. major axis throughout the nebula, but each envelope 2. Shells, which often are found surrounding cores, and farther from the nucleus is more spherical than the previous consist of an extended region of lesser surface brightness envelope. Since PN expansion timescales are usually much which monotonically decreases (often with a linear slope; longer than the sound crossing timescales of the shells, it is Frank,Balick, & Riley 1990) away from the central star out likely that shell-ISM and shell-shell interactions tend to to some maximum radius(r [ 0.1 pc). The shell edges are sphericize PN envelopes over time (Icke 1988). sharp and well deÐned. Halos that are sometimes as much as 104 times fainter 3. Halos, which are larger(r Z 0.1 pc) and more spherical than the bright inner cores are seen, and their detection than cores or shells. Halos are always limb-brightened and requires long integrations using CCDs on a wide patch of often appear as detached rings (partial or complete) sur- sky. High-quality images of PN halos are rare. There exist a rounding (but not generally centered on) the nucleus and few studies of the halos surrounding individual PNe, includ- shell. ing the kinematic analysis of NGC 6826(Bryce, Meaburn, & Walsh1992a), NGC 6543 (Bryce et al. 1992b), and the Compared to the nomenclature ofStanghellini & Pas- chemical analysis of NGC 6751(Chu et al. 1991), NGC quali(1995), for example, their attached halos correspond to 6543(Manchado & Pottasch 1989), and M1-46 (Guerrero et our shells, and their detached halos correspond to our halos. al.1996). However, the dearth of reliable data for PN halos Recently published narrowband CCD images of several has limited previous studies of the group as a whole. PNe(Balick 1987; Balick et al. 1992; Chu, Jacoby, & The goal of this paper is to explore the qualities of PN multiple halos in order to gain insight into the evolution of their parent nuclei. A wealth of information is contained in 1 United States Naval Observatory, USNO/NRL Optical Interfero- meter Project, 3450 Massachusetts Avenue NW, Washington, DC 20392- the brightness distribution (albeit faint) surrounding PN 5420. central stars, including a history of the mass loss (which is 2 Astronomy Department, University of Minnesota, Minneapolis, MN inaccessible by other means). 55455. Some of the properties of and correlations between PN 3 Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195. halo and their central stars have been presented by 4 Department of Astronomy and NAIC, Cornell University, Ithaca, NY Stanghellini& Pasquali (1995) and Frank, van der Veen, & 14853. Balick(1994) using small numbers of observations. In this 226 TIMESCALE CORRELATION METHOD 227

TABLE 1 OBSERVED OBJECTS

T ([O III]) T (Ha ] [N II]) Name R.A. (1950) Decl. (1950)int (s)int (s)

IC289 ...... 0306 16.4 ]61 07 39 1560 1200 NGC 3587 ...... 1111 53.3 ]55 17 21 1260 1020 NGC 2610 ...... 0831 05.0 [15 58 39 600 780 NGC 2438 ...... 0739 32.5 [14 36 56 960 660 NGC 2022 ...... 0539 22.0 ]09 03 54 930 1020 NGC 1514 ...... 0406 08.3 ]30 38 43 210 360 paper, we combine their catalogs with our own obser- The observations and data reduction for the CCD images vations. Though the number of elliptical PNe with halos in are described in° 2, followed by a brief discussion of the each sample is small, we utilize the improved statistics of the individual results in° 3. In° 4, we discuss timescale corre- combined set to analyze the relationships between PN lations between the PN shells and their associated central envelopes and their parent nuclei. nuclei, and we present and evaluate the timescale corre-

FIG. 1.ÈDeep images of selected PNe obtained at the Palomar 5 m telescope using the COSMIC camera. These images have been taken in the light of [O III] j5007 except for NGC 1514 which was taken in the light of [N II] j6584 ] Ha. The images of some PNe (NGC 1514, NGC 2438, and NGC 3587) have been convolved to a resolution of1A.12 and the remainder (IC 289, NGC 2022, and NGC 2610) were left at their original resolution of0A.56. The inner cores of the PNe are shown as the log of the intensity. When visible (NGC 1514, NGC 2022, NGC 2438, and NGC 3587), outer halos are represented as high-contrast linear functions of intensity. 228 HAJIAN ET AL. Vol. 477 lation method which we use to compute distances to PNe Images of NGC 1514 have been previously published by with detached halos. We end with a brief summary of our Chuet al. (1987) using the Palomar Sky Survey (red) plates conclusions in ° 5. and image-tube camera plates ([O III]). The latter does reveal a possible Ðlamentary halo structure surrounding the 2. OBSERVATIONS bright core, which we conÐrm. Our images also reveal two The observations were conducted at the Palomar 5 m limb-brightened regions. It is possible that these bright telescope equipped with the COSMIC system during the regions correspond to the equatorially brightened portions evenings of 1995 December 25È27. The seeing throughout of the ““ halo ÏÏ outÑow, which is bounded by an even fainter the observations was very good(D1A.0È1A.4) although the (and not currently detectable) envelope from the red giant sky transparency was slightly diminished by intermittent progenitor. The physical nature of this complex and and light cirrus clouds. The COSMIC imaging unusual structure must remain speculative until higher spectrometer/camera was used in the imaging mode, and quality images of the halo of NGC 1514 become available. on-chip 2 ] 2 binning yielded 1024 ] 1024 pixel images 3.3. NGC 2022 with a scale of0A.56 per pixel. Images were obtained through [O III] and Ha ] [N II] Ðlters with bandpasses of 30Ó and NGC 2022 is a triple-shell system with a detached halo 100Ó, respectively. The six observed PNe and associated surrounding a bright core/shell structure(Chu 1987; Chu et integration times are listed inTable 1. Short integrations al.1987). While the Ñux in some published images drops to were used to minimize saturation of the nebular cores which the sky background level between the outer edge of the halo are extremely bright in some cases. The resulting images and the outer edge of the shell, our sensitivity is high were calibrated and all frames for each PN and Ðlter were enough to reveal a continuous brightness distribution co-added using standard IRAF routines. During this throughout the nebular system. The bright arc in the halo is process, some of the images were rebinned to1A.12 in order reminiscent of similar features in the halos of PNe which are to improve the image quality without substantially degrad- interacting with their surroundings(Borkowski, Sarazin, & ing the appearance of sharp features, and cosmic rays and Soker1990). However, unlike many PNe which demon- the sky background were removed. The resulting images are strate such interactions(Stanghellini & Pasquali 1995), the displayed in Figure 1. center of the halo of NGC 2022 is not signiÐcantly o†set from the nucleus. 3. RESULTS FOR INDIVIDUAL PNe 3.4. NGC 2438 Though each of the target objects in this study has been NGC 2438 is morphologically similar to NGC 6720 and previously observed (seeBalick 1987; Chu et al. 1987), our IC 1747. The inner core/shell structure is surrounded by a images are the deepest. Three PNe with large halos, NGC rounder halo that shows signiÐcant limb brightening. The 2022, NGC 2438, and NGC 3587, are clearly evident in our halo is elliptical along the same axes as the core, although sample. The amorphous, possibly limb-brightened halo more circular. But because of the nonuniformity of the halo around NGC 1514 is seen in much detail. The bright cores edges, it is not possible to determine if the nucleus and the of IC 289 and NGC 2610 do not show any evidence of geometric center of the nebula are o†set. Our image high- extended halo structure down to our noise level, though we lights the structure in the halo, which is not apparent in do Ðnd a faint shell in each case. Below we make speciÐc earlier images of NGC 2438(Chu et al. 1987; Balick 1987). comments for the individual nebulae, which apply for either The halo is much brighter in [O III] than in Ha ] [N II], the [O III] or the Ha ] [N II] images (i.e., the Ha ] [N II] and our images show that the halo is not completely images appear much the same as the [O III] images unless smooth, but contains streaked features that point radially otherwise noted). from the nucleus. 3.1. IC 289 3.5. NGC 2610 IC 289 appears as a bright elliptical shell, with a major Similar in appearance to IC 289, NGC 2610 is an ellip- axis of length 32A and an axial ratio of 1.46. A faint shell is tical PN with a major axis of extent 34A and an axial ratio of present in our images and in the lower sensitivity images of 1.23. Our image is somewhat deeper than that ofChu et al. Chuet al. (1987) and Balick (1987). The edges of the shell (1987)and Balick (1987), but we Ðnd no new features. are sharp, suggesting that the gas is being conÐned by ram pressure, perhaps by an unseen giant halo. Otherwise, 3.6. NGC 3587 despite our images of IC 289 being the deepest yet obtained, The halo surrounding NGC 3587 has only recently been we Ðnd no direct evidence of limb brightening or associated detected(Kwitter, Chu, & Downes 1991), and published halo structures. information on this beautiful object is scant. In fact, the only other mention of this halo is a spectroscopic report by 3.2. NGC 1514 Manchadoet al. (1992) who measured the 27È39 km s~1 Our [O III] image of NGC 1514 shows a lumpy nebula expansion velocity of the main nebula but were unable to composed of numerous small bubbles, somewhat similar in resolve the >10 km s~1 expansion velocity of the halo. appearance to the inner structure of NGC 6543, and remi- Our images indicate that the halo is more prominent in niscent of the Owl nebula NGC 3587 (see below). The pres- [O III] than in Ha ] [N II] is dramatically o†-center with ence of bubbles at the very edge of the core nebula hints at respect to the nucleus and main nebula, and has a low the presence of halo material which (1) is being swept up eccentricity. InKwitter et al. (1991), only the bright rim of and (2) provides pressure conÐnement. In fact, we have the halo is clearly visible (i.e., the part of the halo that is detected the presence of an amorphous halo surrounding closest to the nucleus). Our observations reveal the opposite the core. half of the detached halo, and we can also identify the frag- No. 1, 1997 TIMESCALE CORRELATION METHOD 229 mentation and clumpiness with the bright parts of the halo any known interacting PN system, surpassing even NGC edge. The surface brightness drops to nearly the back- 246(Soker, Borkowski, & Sarazin 1991). ground level between the halo edge and the main nebula Finally, we point out that it is conceivable that there along the southern part of the nebula. A conical shadow is exists a binary in the nucleus of the Owl nebula for which visible near the northern edge of the shell and halo. The the observed central star is the primary. The period implied shadow emanates from a low-ionization knot just inside the by a 0.06 pc major axis is large, and given by P B 5 ] 105 outer edge of the shell and points back to the central star, (M/M_)0.5 yr, whereM \ Mp ] Ms is the sum of the indicating that an optically thick region is absorbing stellar masses of the primary and (unobserved) secondary. If we photons and shadowing the outer structures. The col- assume that both stars are coeval, then the extreme faint- linearity of the nucleus, the thickest portion of the central ness of the secondary implies that either (1) the main- core, and the shadow suggest that the column density of sequence mass of the primary was smaller than the nebular gas between the shadow and the nucleus is very secondary (which has already passed through the PN phase large. However, the narrow angular size of the shadow and of its evolution and is now a cool and dim white dwarf), or the shape of the shadow edges indicate that a single high- (2) that the secondary is a brown dwarf. While we feel the density region located at the cap of the core outÑow is binary hypothesis of the nucleus of NGC 3587 to be responsible. We would also like to point out that since the unlikely, we cannot completely exclude it based on the recombination timescale(qr D 105[n/cm~3]~1 yr) for the available data. We leave further discussion of the dynamics gas (n D 10 cm~3) is probably smaller than the dynamical of the halo to another paper(Hajian & Frank 1996). timescale of the shell (D6.7 ] 104 yr), the gas within the shadow may have once been ionized and only recently 4. TIMESCALE CORRELATION ANALYSIS recombined. Although the distance between the center of the giant 4.1. T he Sample halo and the observed nucleus is large (20A at a distance of The sample consists of elliptical PNe with large spherical B0.06 pc) and there is distinct edge brightening on the half halos from this paper (NGC 2022, NGC 2438, and NGC of the halo nearest the central star, it seems at Ðrst strange 3587),Frank et al. (1994) excepting NGC 6720, which is that the halo does not appear somewhat elliptical, with the probably a bipolar PN seen in projection(Bryce, Balick, & major axis normal to the direction of the proper motion of Meaburn1994), and Stanghellini & Pasquali (1995). Our the central star. However, models of PN-ISM interaction by Ðnal sample of PNe, the angular radii and expansion veloc- Borkowskiet al. (1990) predict that interacting halos ity of the shells and halos,hi, ho, vi, andvo, the Shklovsky demonstrate signiÐcant ellipticities only at late times in distance and corresponding nuclear luminosity,d and L o, their evolution. Given the dynamical and interpulse time- are listed inTable 2. Visual inspection of publishedSH CCD scales associated with NGC 3587 (see° 4), it would seem images giveshi andho (this paper;Balick et al. 1992; that the giant halo of NGC 3587 has exhausted more than Schwarzet al. 1992). Values ofvi are readily available in the one-quarter of its allotted lifetime as a spherical entity. literature(Weinberger 1989), but due to the faintness of We mention in passing that the displacement between the most halos, published values ofvo are not always available. center of the halo and the nucleus is fractionally larger than The exceptions are the bright halos surrounding NGC 2022

TABLE 2 DATA FROM THE LITERATURE

h h v v d L i o i o SH o Name (A)(A) (km s~1) (km s~1) (kpc) (L _) NGC 2022 ...... 15.4a 38.4a 26b 20c 2.19d 2228.d NGC 2438 ...... 76a 134a 22.4b 7.5c 1.08d 192.d NGC 3587 ...... 196a 278a 29b 5/10 0.58e 46.e NGC 7662 ...... 16f 72f 20g 22g 2.04h 7800.h NGC 6826 ...... 21f 68f 10c 14i 1.54h \2100.h NGC 6543 ...... 40f 170f 17j 7j 1.46h \3000.h NGC 6751 ...... 24f 44f 10k 7k 1.77k 3622.k CN1-5...... 4.2l 19l 18b 5/10 4.63l 1267.l M1-46 ...... 6.8l 29.4l 19.0m 8.0m 4.4m 5370.m NGC 2867 ...... 8.1l 38.5l 18.5b 5/10 2.32l 2710.l NGC 6629 ...... 8.3l 20.8l 6b 5/10 2.14l 6637.l NGC 6751 ...... 13.2l 23.8l 42.5k 10k 1.77k 3622.k TC1 ...... 6.5l 26l 7.5b 5/10 3.04l \14723.l

a This paper. b Weinberger 1989. c Chu& Jacoby 1989. d Stanghellini,Corradi, & Schwarz 1993. e Kaler 1990. f Franket al. 1994. g Balicket al. 1987. h Shaw& Kaler 1985. i Bryceet al. 1992a. j Bryceet al. 1992. k Chuet al. 1991. l Stanghellini& Pasquali 1995. m Guerreroet al. 1996. 230 HAJIAN ET AL. Vol. 477

(Chu& Jacoby 1989), NGC 2438 (Chu & Jacoby 1989), luminosity equation. Using the most recent evolutionary NGC 7662(Balick, Preston, & Icke 1987), NGC 6826 models ofVassiliadis & Wood (1994), with a metallicity (Bryceet al. 1992a), NGC 6543 (Bryce et al. 1992b), NGC Z \ 0.016 appropriate to PNe leads to an expression for q 6751(Chu et al. 1991), and M 1-46 (Guerrero et al. 1996). (Franket al. 1994): ip For the remaining PNe, we consider the cases of assumed L outer halo expansion velocities ofvo \ 5 km s~1 and vo \ log (q ) \ 5.19 [ o . (2) 10 km s~1. This is the strongest assumption of our analysis ip 1.78 ] 104 (but see theAppendix) and underscores the need for high- InTable 3 we presentq andq calculated for each nebula resolution spectroscopy of PNe halo structures. Values of in our sample and inFigureip 2dyn we plot these values against d L ando were selected from the literature(Kaler, Shaw, & each other,q versus q . KwitterSH 1990; Shaw & Kaler 1985; Stanghellini & Pasquali ip dyn 1995; Chuet al. 1991; Guerrero et al. 1996). L o was derived 4.3. T he T imescale Correlation Method by computing a helium Zanstra temperature, then scaling a There is a considerable amount of scatter in Figure 2 model spectrum until the observed colors (e.g.,m and m ) V B from the relationshipq \ q . Both timescales in equa- are achieved. This yields the stellar Ñux which was scaled to tions(1) and (2) areip strongdyn functions of the distance L the adopted distance and results ino. Since the relevant employed in the calculations:q scales linearly with d observable is the Ñux and not the luminosity, we list the dyn values ofd that correspond to the calculation forL o. In this way, weSH avoid biasing the data (this point is further elucidated in° 4.3 and the Appendix). Some PNe nuclei lack detectable helium j4686 Ñuxes, and hydrogen Zanstra temperatures were used in the computation, yielding an upper limit toL o (see, for example,Gathier & Pottasch 1989). 4.2. Nuclear Interpulse and Nebular Dynamical Timescales We begin by assuming that both the halos and shells of individual PNe are created during successive periods of enhanced mass loss each of which is terminated by a thermal pulse(Frank et al. 1994). We then compare the dynamical timescale derived from images and kinematic data with the theoretical interpulse timescales derived from computational modeling. If we treat each shell as expanding with constant velocity, the dynamical timescale between successive envelopes, q , is given by dyn Ah h B q \ o [ i d , (1) dyn vo vi sh where the subscript i and o refer to the inner shell and outer halo, respectively. FIG. 2.ÈPlot of the dynamical timescale between envelopes,q , vs. The interpulse timescale,q , can be found from theoreti- the theoretical helium interpulse timescale of the nucleus,q . Wedyn believe ip cal model calculations (e.g.,Paczynip ski 1970). These depend that the deviations in the data (XÏs) from the solid line, which denotes the relationq \ q , are mostly caused by errors in the Shklovsky distance, on the stellar core mass,Mc, and can be combined with the halo expansionip dyn velocity, and the assumption of zero envelope acceler- relations linkingMc to the plateau luminosity of the core ation. Here we have assumed velocities of 5 km s~1 for those sources (Paczyn ski 1975) to yield a theoretical interpulse timescale without measured halo expansion velocities.

TABLE 3 TIMESCALES FOR PLANETARY NEBULA HALOS

q q q(5) q(10) d (5) d (10) ip dyn TC TC Name (104 yr) (104 yr) (104 yr) (104 yr) (kpc) (kpc) NGC 2022 ...... 11.6 1.30 3.08 . . . 5.18 ^ 1.14 . . . NGC 2438 ...... 15.1 3.48 11.7 . . . 3.62 ^ 0.93 . . . NGC 3587 ...... 15.4 6.66 15.0 13.5 1.38 ^ 0.84 2.88 ^ 1.73 NGC 7662 ...... 5.65 2.25 2.91 . . . 2.63 ^ 0.57 . . . NGC 6826 ...... 11.8 2.54 5.13 . . . 3.11 ^ 0.67 . . . NGC 6543 ...... 10.5 14.3 11.9 . . . 1.21 ^ 0.28 . . . NGC 6751 ...... 9.69 3.07 4.84 . . . 2.79 ^ 0.65 . . . CN1-5...... 13.1 7.38 10.9 7.36 6.82 ^ 2.54 9.88 ^ 3.00 M1-46 ...... 7.74 6.52 6.99 . . . 4.71 ^ 0.95 . . . NGC 2867 ...... 10.9 7.53 9.20 5.88 2.83 ^ 0.92 3.86 ^ 1.07 NGC 6629 ...... 6.56 2.66 3.50 1.16 2.82 ^ 0.81 3.72 ^ 1.31 NGC 6751 ...... 9.69 1.64 3.05 . . . 3.30 ^ 0.69 . . . TC1 ...... 2.31 5.89 4.67 2.35 2.41 ^ 0.65 3.03 ^ 0.82 No. 1, 1997 TIMESCALE CORRELATION METHOD 231 whileq goes as10d2. The Shklovsky distance indicator is a computationallyip simple algorithm that relies on the assumption of a constant ionized nebular mass common to all PNe. Though Ðne for large statistical surveys, Shklovsky NGC 3587 distances are uncertain by a factor ofZ2 for individual PNe (Daub 1982; Cahn,Kaler, & Stanghellini 1992; Terzian 0.9 kpc 1993). In our case, the deviation in the computed values of 1.2 kpc (q , q ) could be explained as the result of errors in the dyn ip 3.0 kpc Shklovsky distance for individual sources. We can uniquely 4.2 kpc derive a distance, which we name the timescale correlation distance,d , by forcing each PN halo to lie exactly along TC the relationship between the two timescales. Equations (1) NGC 2438 and(2) adopt Shklovsky distances to computeq and q . The timescale correlation method forcesq \ qdyn and thusip 4.5 kpc calculates d . dyn ip We beginTC by writing the general form of the equation (1) as 6.0 kpc NGC 2022 Ah h B q (d) \ o [ i d , (3) dyn vo vi where d is the distance to the PN and which we are treating as a free parameter. We note that since L (d) P d2 and L \ FIG. 3.ÈPlots ofq (d) vs.q (d) for selected PNe using eq. (6). The o dyn ip L (d \ d ): separation between the symbols represents * d \ 300 pc. The solid line lies SH alongq \ q , and the intersection between this line and the plotted dyn ip d2 curve for each PN is howd is computed. Numerical values of d are L(d)\L \ k d2 (4) shown. TC o d2 SH and 4.4. Evaluation of Timescale Correlation Method L (d) k It is immediately apparent (see theAppendix) that the log q (d) \ 5.19 [ \ 5.19 [ d2 . 10 ip 1.78 ] 104 1.78 ] 104 fractional uncertainty ind is small and not strongly sensi- tive to uncertainties in theTC observables. Nevertheless, the (5) derivation ford andp is accurate insofar as the TC dTC Equations(3) and (5) are parametric equations in d which envelopes are known to be created in association with deÐne the allowable[(q (d), q (d)] locus for each PN. thermal pulses, the pulses have been correctly modeled, the Mathematically, this curvedyn is givenip by combining the two interacting stellar winds model applies, and q \ q . dyn ip equations to yield Among our sample, the common occurrence of partially disrupted, displaced, and the asymmetrical brightening of C k Ah h B~2D limb-brightened halos suggests that the interacting stellar log q (d) \ 5.19 [ o [ i [q (d)]2 . 10 ip 1.78 ] 104 vo vi dyn winds model applies. The majority of the data in Figure 2 show thatq (d ) andq (d ) are within a factor of 5 of (6) each other.dyn TheSH strong dependenceip SH of both timescales on We plot this relation in the(q , q ) plane in Figure 3 for a the distance used and the large uncertainty in the Shklovsky few PNe. dyn ip distance makes it plausible that the scatter inFigure 2 is the We now impose the constraint result of the uncertainty in the distance. Furthermore, PN central stars lack inherent physical timescales of D104È105 q (d \ d ) \ q (d \ d ) \ q . (7) yr. These arguments lead us to conclude that the data dyn TC ip TC support theq \ q relation. Second order terms such as In other words, there exists a value ofd \ d such that dyn ip equations(6) and (7) are both satisÐed, leaving usTC with the deceleration of the halo and the luminosity evolution of the central star with respect to time have been neglected in our analysis. The Ðrst of these concerns, the halo deceler- C k Aho hiB~2D log q \ 5.19 [ [ q2 . (8) ation, can be dismissed until the dynamical timescale of the 10 1.78 ] 104 v v o i halo is close to the hydrodynamical stopping time, q . Given the observables k,h , v , h , andv , equation (8) can Unfortunately,q is a complicated function of the speedstop of i i o o stop be rapidly solved for q using the Newton-Raphson formal- the central star with respect to the ISM,vo, and the density ism, and we can solve uniquely ford using equation (3) of the ambient medium. A detailed model is necessary to andequation (7). Values of q , q , qTC, andd for each of accurately compute the magnitude of the deceleration for the program nebulae are listeddyn inTableip 3. TheTC comparison each PN. Even so, the results may be skewed by the lack of betweend andd is shown inFigure 4a for assumed knowledge concerning the mass distribution of the ISM SH TC values ofvo \ 5 km s~1 and in Figure 4b for assumed values (Borkowskiet al. 1990; Chu et al. 1991). For the purposes of ofvo \ 10 km s~1. In most cases, we Ðnd that Shklovsky this study, we point out that there are no increasing devi- distances are smaller than timescale correlation distances ations fromq \ q inFigure 4 for large values of q, which (d /d \ 1.58 ^ 0.70), though we await a larger sample we would expectdyn ifip signiÐcant decelerations plague the beforeTC SH investigating possible trends in more detail. sample. 232 HAJIAN ET AL. Vol. 477

FIG.4a FIG.4b FIG. 4.ÈComparison of the Shklovsky distance,d and the timescale correlation distance,d , for the PNe in our master sample. Comparison (a) assuming 5 km s~1 for sources without measured halo expansionSH velocities; (b) assuming expansion velocitiesTC of 10 km s~1.

Our method is strongly tested in the case of the triple- & Masson1996). Though accurate, these techniques are envelope PN NGC 6751, and we are encouraged by the conducive to the analysis of individual PNe. Our analysis results. The two values ofd (one for each pair of has shown that the timescale correlation method can be envelopes) have overlapping 1 TCp error bars. The distance used to derive accurate, rapid, and rigorous distances to a computed using the inner haloÈouter halo pair is smaller large sample of PNe with detached halos. than the distance computed using the coreÈinner halo pair. The largest source of error in the computation ofd is Presumably, this is the result of the deceleration of the halo the expansion velocity of the halo. We would feel moreTC of NGC 6751, which was detected in the echellograms comfortable if measured values ofvo were used in the calcu- published byChu et al. (1991). We compute thatq to lation ofd (andp ), though leaving a large margin of NGC 6751 is 3.30 kpc for the inner haloÈouter halo pairTC if error forv TC results inTC a small penalty forp . Nevertheless, o TC we adoptvo \ 9.9 km s~1 for the outer halo, which corre- we hope that expansion velocities are spectroscopically sponds to a braking of 2.8 km s~1 during the dynamical determined for a large number of PNe with detached halos evolution of the outer halo. Thus, even when considering in the future. Formally, we Ðnd that the average value of the the cases of a halo decelerated from 10 km s~1 to D7km uncertainty ind is 22% for PNe with known observables, TC s~1 and a halo with no deceleration, the timescale corre- and 37% for PNe with assumed values of vo. lation method predicts distances with overlapping 1 p error bars. The second e†ect which we must consider is the lumi- 5. SUMMARY AND CONCLUSIONS nosity evolution of the nucleus. The appropriate value of L o We summarize our conclusions below. in equations(4) and (5) is the luminosity that the central star possesses when it leaves the AGB and begins to travel hori- 1. We have obtained high-quality, deep CCD images in zontally across the H-R diagram. The discrepancy between the light of [O III] and [N II] ] Ha of three PNe with this plateau luminosity,L , and the current luminosity of detached halos, two PNe with attached halos, and one PN the nucleus is the primary sourcepl of the 50% adopted uncer- with an amorphous halo. tainty in k. 2. By combining the data for the three PNe with Although the timescale correlation method is only slight- detached halos with the others inFrank et al. (1994) and ly more complicated than the Shklovsky method, it pos- Stanghellini& Pasquali (1995), we have compiled a list of 13 sesses a higher degree of accuracy. Previous to the pairs of PNe halos/shells. This is the largest list of detached development of the timescale correlation method, the only halo PNe analyzed to date. means available for achieving such accurate distances has 3. We present the timescale correlation method as a been via parallax of a few nearby central stars (Cudworth means for determining distances to PNe with detached 1974; Pieret al. 1993), detailed spectroscopic modeling halos. Our analysis indicates thatd is computationally which has been applied to a few objects(Mendez, Kudritzki, simple and lends itself to large sourceTC samples, while pos- & Simon1985; Mendez et al. 1988), or via expansion paral- sessing relatively high accuracy. The typical uncertainty in lax techniques which have been used to obtain distances to d is B20% for PNe with known values of k, h, and v, and about a dozen PNe (Hajian, Terzian, & Bignell 1993, 1995; risesTC to only B40% in the case of PNe with assumed halo Hajian& Terzian 1996; Masson 1989a, 1989b; Kawamura expansion velocities (and large adopted errors). No. 1, 1997 TIMESCALE CORRELATION METHOD 233

TABLE 4 ERROR ANALYSIS FOR d TC Name c v c v c v c v c v p /d k k hi hi vi vi ho ho vo vo dTC TC NGC 2022 ...... 0.192 0.011 0.023 0.036 0.073 0.219 NGC 2438 ...... 0.092 0.016 0.032 0.084 0.168 0.256 NGC 3587 ...... 0.015 0.014 0.028 0.113 0.567 0.611 NGC 7662 ...... 0.194 0.008 0.016 0.033 0.065 0.215 NGC 6826 ...... 0.174 0.011 0.021 0.044 0.087 0.214 NGC 6543 ...... 0.089 0.007 0.015 0.077 0.154 0.235 NGC 6751 ...... 0.177 0.020 0.040 0.052 0.104 0.231 CN1-5...... 0.106 0.004 0.008 0.066 0.332 0.372 M1-46 ...... 0.156 0.004 0.009 0.046 0.091 0.202 NGC 2867 ...... 0.130 0.003 0.006 0.055 0.276 0.325 NGC 6629 ...... 0.189 0.013 0.027 0.040 0.201 0.288 NGC 6751 ...... 0.193 0.004 0.008 0.029 0.058 0.209 TC1 ...... 0.178 0.006 0.013 0.038 0.189 0.270

NOTE.Èc \ p /k \ 0.5. c \ p /h \ 0.1. c \ p /v \ 0.2. c \ p /h \ 0.1. k hi i vi i ho o c \ p /v \1 0.2 (for known values2 ofv ). c \ p3 /v \ 0.5 (for assumed4 values of 5 vo o o 5 vo o vo).

APPENDIX

We compute the uncertainty in the statisticd , pd , which is due to the contributions from k, h, and v, by Ðrst invoking equation (7),then evaluating equations (3) and (5)TC atd \TC d . Eliminating q from these equations gives TC C Ah h B D k log 4.47 o [ i d \ 5.19 [ d2 , (9) 10 vo vi TC 1.78 ] 104 TC wherek \ L o/d2 , L o is inL _, d and d are in pc,ho andhi are in arcsecs, andvi andvo are in km s~1. Di†erentiating equation (9) with respectSH to the appropriateSH (and uncorrelated) variables gives the error equation: AddB2 AL d B2 AL d B2 AL d B2 AL d B2 p2\d p2 TC ] p2i TC ] p2vi TC ] p2o TC ] p2vo TC , (10) TC k Lk h Lhi Lvi h Lho Lvo which we write in fractional units: Ap B2 dTC \ (c )2(v )2](c )2(v )2](c )2(v )2](c )2(v )2](c )2(v )2 . (11) d k k hi hi vi vi ho ho vo vo TC Inequation (11), we have deÐned the partial derivatives, L d v \ TC (12) x Lx and the fractional error in the observables, p c \ x . (13) x x Evaluating the partial derivatives yields C 5.19 [ log (q) D v \ , (14) k 10.8 [ 2 log (q)

ho d vvo \ v o \ 1.04 TC , (15) h qvo[5.41 [ log (q)] hi d vvi \ v i \ 1.04 TC , (16) h qvi[5.41 [ log (q)] For completeness, we display the contributions from each term to the fractional error ford inTable 4 (when absent,vo is assumed to be 5 km s~1). For the fractional error in the observables, we adoptc \ c \ 0.1,TC c \ c \ 0.2 (for measured hi ho vi vo velocities),c \ 0.5, andcvo \ 0.5 for assumed values ofvo. We would like to emphasize that the large errors inherent to d do not a†ect ourk calculations: according to our algorithm, the relevant quantity is the uncertainty in the Ðducial Ñux, k. SH

We wish to thank the useful suggestions and comments on a draft of this manuscript by Will van der Veen, N. M. Elias, II, and A. Fey. B. B. acknowledges the support of NSF grant AST 94-17112. Y. T. was supported in part by the National 234 HAJIAN ET AL.

Astronomy and Ionosphere Center which is operated by Cornell University under a cooperative management agreement with the National Science Foundation. Observations at Palomar Observatory were made as part of a continuing collaborative agreement between the California Institute of Technology and Cornell University.

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