THE ASTRONOMICAL JOURNAL, 119:2910-2918, 2000 June
© 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

We compare the morphology of the cometary knots in the Helix Nebula in different emission lines using Hubble Space Telescope Wide Field Planetary Camera 2 and Space Telescope Imaging Spectrograph observations. We find that the [N II] 6658 Å line emission from the cometary heads is displaced with respect to H, peaking at a position that is closer to the central ionizing star. This result seems at first sight to be in conflict with simple photoionization models, which predict that the [N II] emission is closer to the ionization front (IF) because it is confined to a thin H⁺-He⁰ layer, a prediction confirmed by calculations with both our own and the CLOUDY programs. However, the ratio of [N II] to H is very temperature sensitive, and the observations can be explained if the knots are modeled as photoevaporating globules. In this case, there is a strong temperature gradient across the IF, resulting in the [N II] emission being depressed in the partially neutral zones. We also find a strong correlation between the strength of the [N II] and [O III] emission in individual knots, with both being higher in the knots that are closer to the central star. On the current evidence, it is unclear whether this is due to interknot variations in metal abundances or in gas temperature.

Key words: planetary nebulae: individual (NGC 7293)

     ¹ Based on observations with the NASAESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.

     The cometary knots (CKs) in the Helix Nebula (NGC 7293), the closest bright planetary nebula (PN), are an integral part of the structure of this object since they contain about the same amount of mass as the ionized (O'Dell & Burkert 1997, hereafter OB) and neutral (Young et al. 1999) components. Numerous theories have been advanced to explain their origin, including the formation of instabilities at the ionization front (Capriotti 1973) and the ionization of preexisting clumps in the PN's neutral envelope, which gains credence from the high degree of clumpiness observed in the molecular shell that surrounds the Helix Nebula (Huggins et al. 1999).

     Whatever their origin, the rather uniform shape of the CKs is primarily determined by the process of photoionization. There is a neutral core of dust and gas photoionized on the side facing the hot central star (which forms a bright cusp of emission on that side), with a fainter, long, limb-brightened, cylindrical tail that points radially away from the central star (O'Dell & Handron 1996). The CKs have rather low ionization, with the [N II] line at 6583 Å being stronger than H and the [O III] emission at 5007 Å being weak compared with the radiation from the main body of the nebula. Indeed, it is the weakness of their 5007 Å emission that allows their dust component to be obvious, since the closer CKs are seen in silhouette against the nebula.

     The Helix Nebula appears to be a thick disk of material showing a progression of ionization states (O'Dell 1998; Henry, Kwitter, & Dufour 1999). The inner parts (radii less than 100) are highly ionized and best visible in He⁺², with the next most visible part being the He⁺ zone, which is best traced by [O III] emission. Outside this is a narrow He⁰ zone, best seen in [N II] emission. The outer boundary of the ionized zones is delineated by the ionization front, which is best traced by [O I] and [S II] emission. Surrounding the ionization front is a neutral and molecular zone, which is well traced by CO and H₂ (Huggins et al. 1999). The fact that the long CO arcs closely match the low-ionization emission-line arcs (Huggins et al. 1999) that give the object its name as the Helix Nebula proves only that the nebula is optically thick to ionizing photons in the plane of the disk, whereas the nebula is probably optically thin in directions close to the polar axis.

     CKs may be common in many planetary nebulae. NGC 6853 shows similar structure on ground-based images (Meaburn & López 1993) and on unpublished Hubble Space Telescope (HST) Wide Field Planetary Camera 2 (WFPC2) program GO 6280 images. More recent publicly released images with WFPC2 also show CKs in the Ring Nebula (NGC 6720) and the in Eskimo Nebula (NGC 2392). The Helix Nebula, at a trigonometric parallax distance of 213 pc (Harris et al. 1997), is closer than any of these objects and is the closest PN with a surface brightness this high. Therefore, it is important to understand the origin and mechanisms of the CKs in the Helix Nebula, because the same processes are probably occurring in a significant fraction of the PNs.

     The correctness of the basic model, that these are small neutral knots of about 3 × 10^-5 M (OB) photoionized on the starward side, seems established from the observed variation of the cusp surface brightness in H with angular distance from the central star (O'Dell & Handron 1996). If all ionizing photons were used up in balancing the recombinations in the ionized gas surrounding the CKs, then one would expect their surface brightness in a given recombination line to vary linearly with the arriving flux of ionizing photons, with the constant of proportionality being equal to the ratio of the line emission coefficient to the recombination coefficient. Although the surface brightnesses of the CKs are observed to drop with increasing distance from the central star, the general level of the H surface brightnesses is about a factor of 2 less than expected from this simple model, given the total luminosity of the nebula. This discrepancy would be even greater if the main body of the nebula were optically thin to ionizing radiation (density bounded) in some directions, which is probably the case.

     Such a discrepancy should not be surprising, however, since the density in the ionized envelopes of the CKs is low enough that a substantial fraction of the stellar ionizing photons can reach the ionization front to ionize fresh neutral gas. In this case, the rate of recombinations per unit area in the ionized envelope can be substantially lower than the incident ionizing photon flux, with the remainder of the photon flux being balanced by the flux of fresh gas particles through the ionization front (IF). Indeed, detailed models of this process (López-Martín et al. 2000) show that the observed H surface brightnesses can be well fitted by an ionizing photon luminosity for the central star of Q = (35) × 10⁴⁵ s^-1. This is fully consistent with the most recent determination of the integrated brightness of the nebula (O'Dell 1998), which implies Q = 4 × 10⁴⁵ s^-1.

     In this paper, we will show that there is a second peculiarity of the emission from the CKs that runs contrary to the predictions of the simplest models. Again, however, we find that a slightly more sophisticated model can largely explain the observations.

     This paper draws on both new observations and old HST observations. The new observations are Space Telescope Imaging Spectrograph (STIS) spectra obtained on 1998 November 18 as part of program GO 7286. The spectrum used here is a wide (2) slit exposure of 600 s centered on the H line. CK 378-801 was centered on the slit in position angle 356°, which was selected to lie along a line from the central star passing through the center of the CK. Since the chord size of the cusp of CK 378-801 was about equal to 18, the result was essentially a slitless spectrum that included both the H line and the two members of the [N II] doublet that straddles it, at 6548 and 6583 Å, and is shown in Figure 1. Throughout this paper we will use the coordinate-based designation system introduced in OB, thus avoiding the problems of serial catalogs, which must be revised with improved surveys. The image in Figure 1 has been calibrated in flux units, and the tilt of the spectrum has been corrected. A notable feature of this spectrum is the displacement of the H emission, which is closer to the center of the CK than the [N II] emission. This is a counterintuitive result within the framework of simple photoionization theory. H emission should arise from all the regions of ionized gas, but [N II] should come from only the H⁺-He⁰ zone that lies closest to the CK's ionization front. This unexpected relative distribution was confirmed more quantitatively by plotting the surface brightness as a function of distance from the center of the CK, averaged over a sector of 50° centered on the CKcentral star line. It was found that the H emission peaked at 2.2 STIS pixels (005 pixel^-1) closer to the center of the CK than the [N II] emission.

FIG. 1.Slitless spectrum of CK 378-801 obtained with STIS. The central cusp is of H at 6563 Å and is straddled by the two [N II] lines at 6548 Å (left) and 6583 Å. One can see that while the H line is strongest in the nebular background (central vertical strip), the [N II] 6583 Å line is much stronger in the cusp. It is also apparent that the H cusp emission is farther away from the central star that causes the ionization (off the image toward the bottom) than the [N II] emission.

     This unanticipated result caused us to reexamine the earlier WFPC2 images, which contain several hundred CKs, to see if a similar result is found. The data set employed was the combined WFPC2 images from programs GTO 5086 and GO 5311 calibrated in H, [N II] 6583 Å, and [O III] 5007 Å emission-line surface brightnesses according to the technique described in O'Dell & Handron (1996). The combined images are shown in Habing & Lamers (1997). Based on the absence of overlapping CKs and the presence of smooth adjacent emission from the main part of the nebula, eight CKs were selected for study. In addition to CK 378-801, the objects included were CK 352-815, 358-800, 360-751, 386-749, 398-752, 428-900, and 473-931. These objects are all very similar, with an average chord size of 156 ± 026. The angular separation of the knots from the central star ranges from 95 to 147, corresponding to a projected physical separation ranging from 0.1 to 0.15 pc. The true physical separation from the central star is probably not much larger than this, placing these CKs inside or at the boundary of the inner He⁺² zone of the nebula (O'Dell 1998). Samples about 10 square centered on each object were taken. The CK centers were identified and then the averaged surface brightnesses were again determined as a function of distance from these centers. The sector widths were always about 54° and centered on a line along the CKcentral star direction.

     Table 1 shows the observed background-subtracted cusp intensities, I_c, of the individual knots in each of the three emission lines, together with the intensity of the nebula background at the position of each knot, I. Most knots show cusps that are brighter in [N II] than in H, although the background nebula is always brighter in H. All the knots are very faint in [O III], and in two cases there was no discernible emission in this line. All except one of the knots show dark cores in [O III], the intensity of which is listed in the table as I_d. The intensity-weighted mean radius, r, for each knot in each emission line is also shown in the table. The last two rows of the table show the ensemble average of each column over the eight CKs, together with the rms deviation from this average.

TABLE 1     OBSERVED PROPERTIES OF INDIVIDUAL KNOTS FROM WFC IMAGES

     It can be seen that r is slightly larger for [N II] than for H for seven of the eight CKs (for CK 358-800 they are equal), by an average of about 8% (004). The mean radius for [O III] is nearly twice that for H. It is noteworthy that, although the dispersion in H brightnesses is rather small, the dispersion in [N II] brightnesses is very large, amounting to a factor of 4 difference between the brightest and faintest CKs. We will return to this point in § 4.

     To produce an average CK brightness profile in each of the three lines (shown in Fig. 2), the radial scale of each knot was first normalized by the individual r for H, which has been divided by the ensemble average of the same. Furthermore, the intensity in each profile was normalized by the integrated area under the profile between a minimum and a maximum radius. These radii were chosen to be 02 and 10 for H and [N II], respectively, to cover the bulk of the cusp emission, and 06 and 15 for [O III], to avoid the region affected by extinction from the CK neutral core.

FIG. 2.Composite observed profiles resulting from averaging eight individual cometary knots (see text for details of the averaging process). Top: Emission-line intensity of H (triangles), [N II] 6584 (stars), and [O III] 5007 (circles) as a function of projected radius. Bottom: The ratio of [N II] to H. In all cases the error bars show the rms variation between the scaled, normalized profiles of the individual knots.

     The displacement between the peaks in H and [N II] is quite apparent in the figure. This is best seen in the ratio plot (bottom), where the relative strength of [N II] increases between radii of 05 and 08. Beyond 08, the ratio declines again since the [N II] emission falls off more sharply than H at large radii. The outer decline in the ratio is quite understandable since N⁺ will be ionized to N⁺² at large distances from the IF, but the initial rise is harder to understand. The first ionization potentials of hydrogen and nitrogen are so close that the ratio of N⁺ to N⁰ should be closely linked to the ratio of H⁺ to H⁰ throughout the IF, and furthermore, the peak electron densities reached in the CK cusps (10³ cm^-3) are low compared with the critical density of the 6583 Å line. Hence, the only mechanism that seems capable of explaining the initial rise in the ratio of [N II] to H is temperature variation in the emitting gas. In the following section, we examine this scenario in the context of photoevaporating flow models for the CKs.

     Photoevaporating flows from externally ionized neutral condensations have been extensively studied in the context of H II region cometary globules (e.g., Oort & Spitzer 1955; Dyson 1968; Kahn 1969; Bertoldi 1989; Bertoldi & McKee 1990). More recently, similar models have been applied to photoevaporating disks (proplyds; O'Dell, Wen, & Hu 1993) in the Orion Nebula (see, e.g., Henney et al. 1996; Johnstone, Hollenbach, & Bally 1998; Henney & Arthur 1998; Richling & Yorke 1998). Specific applications of such models to CKs in PNs have also been carried out (Mellema et al. 1998; Cantó et al. 1998; López-Martín et al. 2000).

     We use a generalization of the transonic photoevaporating flow model that has recently been successfully applied to the Orion Nebula proplyds (Henney & Arthur 1998; Henney & O'Dell 1999). In those models, the ionization front (IF) was assumed to be thin compared with its radius of curvature and to be hemispherical on the side facing the ionizing star. The ionized gas was assumed to flow outward at its sound speed from the IF, corresponding to a D-critical IF (Kahn 1954). Pressure gradients in the diverging transonic flow, which was assumed to be isothermal, cause the gas to accelerate, resulting in a density profile that falls faster than r^-2 and which mimics an exponential distribution at small separations from the IF. In the case of the Helix CK, however, the IF thickness, which is a few times the mean free path for ionizing photons, is not small compared with the knot radius, which is of order 2 × 10¹⁵ cm. The mean free path at the Lyman limit is 2 × 10¹⁴ cm for typical cusp densities of 1000 cm^-3, but the effective mean free path is likely to be about 6 times larger than this (López-Martín et al. 2000), since the ionizing radiation field from the central star is so hard (T_* 10⁵ K; Górny, Stasiska, & Tylenda 1997) and the effective absorption coefficient of hydrogen is smaller than that at the edge of the Lyman continuum. Furthermore, the gas in the photoevaporating flow will not be in static thermal equilibrium, since the heating timescale is given by the photoionization timescale of roughly 300 yr, comparable to the dynamic timescale.

     Bertoldi (1989) presented self-consistent dynamic models of the ionization and thermal structure of such photoevaporating flows in which the ionizing source was an O star (much softer than in the Helix case). In this paper, we adopt a more empirical approach in which we use simple parameterizations of the temperature, velocity, and ionization profiles of the flow. For the temperature, we employ a profile that starts at T = T₀ at an inner radius (r = r₀) and increases to T = T at large r, over a characteristic scale H_T:



For the gas velocity, we use a standard -law, as commonly used in stellar wind modeling:



where u₀ is the (subsonic) velocity at the inner boundary, u is the (supersonic) terminal velocity, and controls the steepness of the acceleration (see, e.g., Lamers & Cassinelli 1999). The total gas density in the flow is calculated from the mass continuity equation. For the ionization fractions, we assume that the ratio of the densities of any two successive ionization stages, j - 1 and j, of an element, k, is of the form



where r_jk is the radius at which the two ion fractions are equal and H_jk is the width of the transition zone. Note that this functional form forces the element to be completely neutral at r = r₀. For simplicity, we assume that there are only three ionization transitions in the flow, corresponding to the ionization of H⁰, He⁰, and He⁺. The ionizations of N⁰ and O⁰ are assumed to follow those of H⁰, those of N⁺ and O⁺ to follow He⁰, and those of N⁺² and O⁺² to follow He⁺.

     Although the model as stated has 14 free parameters (including the knot radius and peak density), many of these are not really free since they are strongly constrained by physics and by observations. A reasonable model must have slow-moving, neutral, 10³ K gas at the inner boundary and mildly supersonic, ionized, 10⁴ K gas at large radii. In addition, the thickness of the hydrogen IF should be a few tenths of the globule radius (see above) and the flow should pass through its sonic point in the region in which hydrogen is about half-ionized. To constrain the models further, we choose parameters for the velocity law and H⁰ IF that closely reproduce Bertoldi's model 3 (1989; illustrated in his Figs. 4 and 5). Except for the relatively soft ionizing radiation field, this model has parameters that approximately match those of the Helix CKs. The temperature is likely to be far more sensitive than the ionization to the spectral shape of the ionizing radiation field, so we feel at liberty to modify the temperature profile in the fully ionized region from that of Bertoldi's model. To constrain the temperature profile in this region, we try to reproduce the O⁺² temperature of 11,700 ± 700 K observed by O'Dell (1998).

     The only remaining important free parameters are the position of the He⁰ and He⁺ ionization fronts. These are chosen to give reasonable agreement with the observed emission-line profiles (see next section). The input parameters of this illustrative model are listed in Table 2, together with the calculated mean temperatures of the N⁺ and O⁺² zones. The resulting density, temperature, velocity, and ionization profiles of the model are shown in Figures 3a3d.

TABLE 2     PARAMETERS OF ILLUSTRATIVE PHOTOEVAPORATING FLOW MODEL

FIG. 3.Physical parameters of an illustrative photoevaporating flow model, all as a function of radius in units of the IF radius. (a) Logarithm of the hydrogen number density of the flow, showing the ionized density (solid line), and the total (ionized plus neutral) density (dashed line). (b) Assumed gas temperature in the flow. (c) Flow speed (solid line) and sound speed (dashed line). (d) Ionization fractions of hydrogen (solid lines) and nitrogen and oxygen (dashed lines). (e) Logarithm of line emissivities in photons cm-3 s-1: H (solid line), [N II] 6584 (dashed line), [O III] 5007 (dot-dashed line). (d) Emissivity ratios: [N II] to H (solid line) and [O III] to H (dashed line).

     Figure 4 shows the temperature dependence of the line emission coefficients that we use. The form of the [N II] and [O III] curves is taken from Mellema (1993), but the absolute values have been scaled to agree with the values of G. J. Ferland (1998, private communication) at 10⁴ K reported in O'Dell (1998). The H curve is a power-law fit to the table given by Osterbrock (1989). The curves shown take account of the N and O abundances (using the O'Dell 1998 values) but must be multiplied by the relevant ion fractions and the proton and electron densities to give the photon emissivity in 4 sr (photons cm^-3 s^-1). The radial dependence of these emissivities for the illustrative model are shown in Figure 3e, while the emissivity ratios [N II]/H and [O III]/H are shown in Figure 3f. The initial rise in the ratio of [N II] to H is due to the temperature gradient near the IF, whereas the fall in this ratio at larger radii is due to the decline in the N⁺ fraction. At the same radii for which the ratio of [N II] to H falls, the ratio of [O III] to H rises because of the increase in the O⁺² fraction.

FIG. 4.Temperature dependence of line emission coefficients in photons cm-3 s-1 for H, [N II] 6584, and [O III] 5007, supposing that each element is entirely in the relevant ionization stage.

     To compare the predictions of our model with the observations of the cometary knots, we integrate the line emissivity along different lines of sight to produce a simulated brightness profile of the model in each emission line. For simplicity, we assume that the symmetry axis of the knot lies in the plane of the sky. We also assume that the ionized gas density at the IF declines with angle from the symmetry axis as



and that the radial run of all physical variables in the evaporating flow is the same at all angles from the symmetry axis. The physical motivation for equation (4) is that simple theory (see, e.g., Bertoldi 1989) predicts the limiting cases of n() = n₀ when most ionizing photons are used up in balancing recombinations in the ionized flow and of n() = n₀ cos when most ionizing photons reach the IF to ionize fresh gas. The CKs are probably closer to the latter limit, so that b = 1 is appropriate. The parameter a is then introduced to account for the diffuse ionizing photons that would prevent n() from going to zero at = 90 (López-Martín et al. 2000). We choose a = 0.3 to reproduce the observed form of the drop in H intensity around the cusp (OB). It should be noted that only that part of the brightness profiles at projected radii less than r₀ is at all affected by the form chosen for n().

     Extinction of background emission from the nebula by dust in the neutral core of the CK was accounted for in calculating the model profiles, assuming that the core has an optical depth of 0.35, corresponding to the mean I_d in Table 1. This was an important effect only in the case of the [O III] line. The resultant simulated brightness profiles were convolved with the WFC point-spread function and scaled to give the same mean radius in H as the observed profiles and are shown in Figure 5. The model profiles can be directly compared with the composite observed profile of Figure 2, and it can be seen that the agreement is quite good. In particular, the model reproduces well the magnitude and radial variation of the ratio of [N II] to H. This ratio is roughly constant for r < r₀ and initially rises in the photoevaporating flow, reaching a maximum at about 1.5r₀ and falling thereafter. The model also reproduces quite well the observed behavior of the [O III] emission, which peaks considerably outside the H and [N II] emission but at a much lower level. In the model, this is due to the fact that the O⁺² ion fraction is very small close to the IF and does not become significant until r 2r₀, by which time the density has fallen by about an order of magnitude. Although the observed [O III] intensity seems to fall more rapidly at large radii than is predicted by the model, not much should be read into this, since the data are very noisy there. As a result, the position of the He⁺ IF is not well constrained in the model.

FIG. 5.Simulated model profiles using the parameters of Table 2. The radius of the ionization front, r0, is marked by the vertical line. Top: Emission-line intensity as a function of projected radius for H (solid line), [N II] 6584 (dashed line), and [O III] 5007 (dot-dashed line). Bottom: The ratio of [N II] to H.

     The mean [N II] and [O III] temperatures of the model are 10,300 and 11,900 K, respectively. The latter is consistent with the determination of O'Dell (1998), while the former is about 1000 K higher than his value, 9400 ± 200 K. It should be recalled, however, that O'Dell's temperature measurements were of the background nebula rather than the knots themselves.

     We have shown in § 2 that there exist significant differences between the spatial distribution of the emission in the H and [N II] lines from the heads of cometary knots in the Helix Nebula. At large radii from the center of the knot, the [N II] emission declines more rapidly than H, which is expected since N⁺ is ionized to N⁺² in those regions. However, in the region of the knot cusp, the [N II] emission peaks at a slightly larger radius that the H emission (that is, slightly closer to the ionizing central star of the nebula). This behavior can be seen in both the slitless STIS spectrum (Fig. 1) and in the average WFC brightness profiles (Fig. 2).

     We have demonstrated in § 3.2 that a simple ad hoc model of the ionization and thermal structure of a photoevaporating flow can explain this behavior so long as the increase in temperature between the neutral and ionized gas occurs over a length scale equal to the thickness of the ionization front. Static photoionization models cannot reproduce the observed behavior since they predict that the temperature increases rapidly to near the fully ionized value while the gas is still largely neutral, leading to a roughly constant ratio of [N II] to H in the regions where the emission of these lines is important. We have verified this using the photoionization code CLOUDY (Ferland et al. 1998) to calculate a model that approximates the physical conditions of the Helix knots, finding that the gas temperature reaches 10⁴ K at the point at which H is only 20% ionized. Indeed, the CLOUDY model predicts that the peaks of H and [N II] emission should coincide. We illustrate this in Figure 6 by showing the results of a model that is identical to that described in § 3.2 except that the scale height of the temperature distribution, H_T, has been reduced from 0.15r₀ to 0.03r₀ to approximate the results of the CLOUDY model. It can be seen that the model overpredicts the [N II] intensity and does not show the offset between the H and [N II] peaks seen in both the observations and the model of § 3.2.

FIG. 6.Same as Fig. 5, but for a model in which the temperature gradient is much sharper, leading to a roughly constant temperature in the fully ionized zone. This model fails to reproduce the observed offset between the peaks of the [N II] and H emission.

     The key feature of the successful model is hence the slow increase in temperature as one passes from the neutral to the ionized zones. Physically, this is due to the fact that the gas is moving through the ionization front with a dynamic timescale comparable to the ionization and heating timescales. In such a case, the gas does not have time to adjust to the equilibrium temperature corresponding to each degree of ionization. Adiabatic expansion cooling of the flow may also have an effect (Bertoldi 1989). To verify that this explanation is correct, it is important to carry out fully self-consistent models of the dynamics and ionization of photoevaporating flows in the context of the Helix knots. The closest existing calculations are those of Bertoldi (1989), but these were carried out for much softer ionizing spectra than is appropriate for the Helix and considered only the ionization of hydrogen and helium. Existing photoionization codes such as CLOUDY currently assume static ionization and thermal equilibrium and hence will require substantial modification to model situations in which the flow dynamics are important.

     To improve the signal-to-noise ratio of the observations, our analysis has concentrated on average knot brightness profiles. However, Table 1 shows that there are significant variations between the individual knots of our sample. Although the peak H brightness is almost the same for each knot, the peak [N II] flux varies by a factor of more than 4. The [O III] emission is also highly variable between knots, with some knots showing both emission and absorption, while others are completely invisible in this line. Futhermore, as shown in Figure 7, the ratio of [O III] to H is strongly positively correlated with the ratio of [N II] to H, which declines with increasing projected distance of the knot from the central star.

FIG. 7.(a) Correlation between line ratios [N II]/H and [O III]/H for individual cometary knots, using the cusp intensities from Table 1. Error bars are estimated from the pixel-to-pixel noise of the images. For two knots, no [O III] emission was detectable and so only an upper limit is shown. (b) Variation of the ratio of [N II] to H with projected distance of the cometary knot from the central star.

     Three possible explanations suggest themselves for this interknot variation. First, the gas-phase abundances of nitrogen and oxygen may vary between the knots. Since nitrogen cannot be depleted onto grains at nebular temperatures, this would be a variation in the elemental mix of the knot material itself, with the closer knots being more metal-rich. If this explanation were correct, then it would argue in favor of the knots' being relics of condensations ejected during the central star's asymptotic giant branch phase, rather than having been formed by some mechanism during the evolution of the nebula.

     Second, the degree of ionization may be different for different knots. Between 120 and 200 from the central star, the H surface brightness of the nebula increases by a factor of 2, while the He II 4686 emission falls to almost zero (O'Dell 1998). This probably corresponds to the transition between the He⁺² and He⁺ zones of the nebula and closely coincides with the region of sharply changing line ratios (Fig. 7b). However, the sense of the variation that we see is opposite to what would be expected if the closer knots were subject to a higher flux of photons with energies greater than 54 eV. In such a case, nitrogen and oxygen would reach their triply ionized state sooner, and this may prevent both N⁺ and O⁺² from reaching a significant ionization fraction in the flow. The result would be a reduction in the ratio of [N II] to H and of [O III] to H at shorter distances, which is the opposite of what is observed.

     Third, there may be variations in the mean temperature of the ionized flows from the individual knots. If the closer knots had generally higher temperatures throughout their emitting zones, then this would enhance both the [N II] and the [O III] emission with respect to H because of the temperature dependence of the emissivity (Fig. 4). Such a temperature variation is plausible because of the increased hardness of the ionizing radiation in the He⁺ zone discussed in the previous paragraph. With the current data, it is impossible to judge whether abundance or temperature variations are responsible. To discriminate between the two, it is important to extend the study to a much larger sample of knots, spanning a greater range of distances from the central star. Measurements of temperature-sensitive line ratios in individual knots would also help decide the issue.

     In conclusion, we can say that the cometary knots in the Helix Nebula display an unusual distribution in emission of the recombination line H with respect to the collisionally excited [N II] lines. However, this apparently counterintuitive distribution is satisfactorily explained by a photoevaporative model that has a strong decrease in electron temperature as one approaches the ionization front of the cometary knot. Provocative variations between cometary knots probably contain additional information about the detailed processes that are occurring. These variations are worth pursuing, as only a highly accurate model can determine the rate of mass loss from the cometary knots and hence determine their survival after having been gravitationally decoupled from the central stars.

     This work was conducted in part while C. R. O. was a guest of the Max-Planck-Institut für Astronomie in Heidelberg, Germany, and the Rice University portions were supported by grant GO-07286.01-96A from the Space Telescope Science Institute. W. J. H. acknowledges financial support from DGAPA-UNAM projects IN128698 and IN117799 and from CONACyT project E-25470, Mexico.

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