TANGENTIAL MOTIONS AND SPECTROSCOPY WITHIN NGC 6720, THE RING NEBULA^*

We obtained new Hubble Space Telescope (HST)/WFPC2 images as parts of programs GO 11231 and GO 11232. The GO 11231 observations were in the F469N (He ii), F547M (continuum), and F673N ([S ii]) filters and executed on 2008 April 23. Triple exposures of 60, 200, and 260 s, respectively, were combined to provide high signal-to-noise ratio (S/N) images free from cosmic ray events. The GO 11232 observations were made in the F658N ([N ii]) and F502N ([O iii]) filters on 2008 May 6. Again triple exposures of 260 s were made in each filter and similarly combined using IRAF tasks.⁴ Identical pointing was employed with the central star of the nebula centered in CCD3 of the WFPC2, duplicating as close as possible the conditions of earlier observations in F502N and F658N in program GO 7632, except that the current two-gyro pointing restrictions of the HST caused us to adopt a nearly reciprocal position angle (P.A.) of 2132, while the GO 7632 observations were made with P.A. = 338. The interval between the GO 7632 and GO 11232 observations in the F502N and F658N filters used for astrometry is 9.557 years. The GO 11232 images were aligned with the geomap and geotran IRAF tasks using four stars, including the central star. The alignment of the central star was corrected with the imshift task to an accuracy of better than 002, leaving the possibility of a poorer alignment accuracy in rotation for objects further out in the nebula, but eliminating any effects of the other field stars not moving in the same manner as the central star.

The Ring Nebula has been the target of many spectroscopic programs at various spectral resolutions, the latest and most comprehensive high-resolution study being that of O'Dell et al. (2007b), while recent moderate-resolution studies are those of Guerrero et al. (1997), Garnett & Dinerstein (2001), and Liu et al. (2004). Moderate-resolution spectroscopy offers the opportunity to measure the most useful diagnostic emission lines, but previous studies have not combined the study of the critical position angles (the major axis points toward P.A. = 60° and the minor axis points toward P.A. = 150°) and full wavelength coverage. We have closed this information gap by making new observations with the Boller and Chivens Spectrograph mounted on the 2.12 m telescope at the San Pedro Martir Observatory.

It was expected that these observations would also provide ground-based telescope reference data for calibrating the F469N and F673N filters of the WFPC2, which meant that we used an unusually wide slit (in order to diminish the effects of comparing high and low spatial resolution data). We employed a 300 μm wide entrance slit (393). The scale of the Site CCD was 105 pixel along the slit and 3.05 Å pixel⁻¹ along the dispersion direction. The full width at half-maximum (FWHM) signal for the final combined spectra along the direction of the dispersion was 9.4 Å.

Observations were made at P.A. = 150° on the night of 2008 July 25 and at P.A. = 60° on the night of 2008 July 26 under photometric conditions. The flux standard BD+25 3941 in the IRAF database was used for both nights. Various exposure times were used in order to avoid the problems of the bright lines saturating on the longer exposures. For P.A. = 150° we obtained seven exposures of 180 s, five of 600 s, and four of 1200 s for a total exposure time of 9060 s. For P.A. = 60° we obtained three exposures of 180 s, three of 600 s, and two of 1200 s for a total exposure time of 4740 s. The seeing FWHM was about 23 on both nights. Twilight sky exposures were made on the evening of the P.A. = 150° observations and were used for characterizing the small flat-field corrections.

The data were reduced using IRAF tasks, which included flat-field correction, distortion correction, wavelength calibration, and rendering into cgs system surface brightness units (ergs s⁻¹ cm⁻² Å⁻¹ steradian⁻¹). The spectra were combined to form a single spectrum for each position angle, with only the shorter exposure time observations used for the brightest emission lines. Digital versions of these spectra are available in the electronic form of this paper.

Night-sky lines were present as were the high-ionization lines arising from the outer halo of the Ring Nebula. The regions beyond the bright ring were used to determine how much should be subtracted from the ring spectra. A final pair of background-subtracted spectra of 107 pixels length centered on the bright ring was created. Both of these spectra were sampled in three regions called top (25 pixels), bottom (bot, 25 pixels), and middle (mid, 13 pixels) with the results presented in Table 1. An illustration of the quality of the spectra is shown in Figure 1. The range of pixels measured from the central star, with positive numbers away from the direction of the P.A., is given in Table 2. The pairs of middle samples were adopted because of the much lower surface brightness near the central star. The sample region spectra were measured with IRAF task "splot," with deconvolution of line blends where necessary. Not all features were measured, in particular those blended lines that would not be useful in a quantitative analysis. The interstellar extinction correction was determined from the F(Hα)/F(Hβ) and F(Hδ)/F(Hβ) ratios using the general extinction curve presented in Osterbrock & Ferland (2006) and their theoretical line ratios of 2.87 and 0.256, respectively. The weighted average extinction was c_Hβ = 0.13 ± 0.04 (in close agreement with the value of 0.14 derived by Guerrero et al. 1997 but about half that of Barker 1987) and the same extinction correction was applied to each sample. The extinction curve utilized and the results of this analysis of the samples are presented in Table 1.

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We also derived c_Hβ pixel by pixel along both slits. This indicated that there is a sharp rise just outside the peak in Hβ brightness in the directions P.A. = 150°, 240°, and 330°, but not toward 60°. These rises are like those shown in Garnett & Dinerstein (2001) toward P.A. = 271°, although they questioned the reality of the rise because it might be due to the misalignment of their spectra, which is not an issue with our observations. Since the peaks in brightness occur just inside the projection of the ionization front and the front is curved, the rise in extinction just outside the peaks is probably due to dust in dense neutral gas projected onto ionized material inside the ionization front.

Our spectra are comparable in resolution to the 7 Å resolution spectra of Guerrero et al. (1997) at similar P.A. values and with a slit width of 12, however, their exposure time along the major axis was 1800 s (ours was 4740 s) and the sum of the exposure times of their spectra nearly along the minor axis was 4500 s (ours was 9060 s), although it should be noted that they covered a wider wavelength range (3727–7330 Å). Our resolution was poorer than the 2.1–2.3 Å reported by Garnett & Dinerstein (2001), who had a total exposure time of 3600 s at P.A. = 91° with a 25 wide slit, but covered a wider wavelength range than their 4150–5000 Å sample. The study of Liu et al. (2004) was made with a P.A. = 90° slit at a resolution of 1.5 Å in their blue samples and 3.5 Å in the red, both with a 1'' slit and a maximum total exposure time in their reddest sample of 2560 s. They reduced the long-slit spectrum to a single spectrum, which we show in Section 4.1 to blend regions of quite different conditions. Their total wavelength coverage was 3600–8010 Å. Because of the wider entrance slit and longer exposure times, our data are of much higher spectrophotometric quality than these earlier studies, within the limitations of the spectral resolution. A measure of the large signal in this program is that the 4922 Å line of He i in the southeast peak of the main ring gave an individual 1200 s exposure signal of 850 counts in 1 pixel along the slit, even though this line is only about 1% of the flux of the Hβ line and this observation represents only 13% of the entire signal for the 4922 Å line.

Since the central star has a surprisingly large spread in published V magnitude brightness (15.0–16.2; O'Dell et al. 2007b), we carefully measured this star on our spectra at 5550 Å (near the center of the V bandpass) and determined that the correct value is V = 15.65 ± 0.05, in good agreement with the value of 15.75 determined by Harris et al. (2007). The small difference in our new results and the value adopted in our earlier study of the nebula (O'Dell et al. 2007b) mean that the stellar properties adopted in that paper do not need to be changed.

Our method of comparison was to magnify the first epoch images, align these with the second epoch image in the same filter, normalize them so that the total signal was the same, then take their ratio. We noted that the ratio images began as being highly structured, indicating a mismatch, then became nearly gray in a small scaling range, then again became structured at larger scaling factors. Samples of the ratio images used in the analysis are shown in Figure 2. This rather qualitative method gave a best scaling factor of 1.0016 ± 0.0005 for the F658N images and 1.0019 ± 0.001 for the F502N images, yielding annual expansion rates over the 9.557 year interval of 0.17 ± 0.05 mas yr⁻¹ arcsec⁻¹ and 0.20 ± 0.05 mas yr⁻¹ arcsec⁻¹ respectively, a weighted average (F502N being given one half weight because of less structure in that filter) of 0.18 ± 0.05 mas yr⁻¹ arcsec⁻¹, only a fraction of the annual expansion rates previously derived from segmented images and arguing that those rates are in error because of the use of images divided onto four CCDs. The probable errors of our present study are very qualitative as they depend strictly on the judgment about when the fit is best. Nonhomologous expansion would have manifested itself by different parts of the ratio image becoming featureless at different scale factors. Qualitatively, this did not seem to be the case except in the bright northern quadrant, where some nonhomogeneity could be seen in the inner and outer parts of the brightest section (which was homologous).

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The more quantitative method employed compared the position of isolated individual features (bright in the case of F658N and dark in the case of F502N) on aligned, normalized images. The method is one of shifting small sections of an image and identifying when the differences are least, a method previously developed (Hartigan et al. 2001) for measurement of Herbig–Haro features. The results are shown in Figure 3 and are presented in Tables 3 and 4. In the same manner as the comparison of images method, we can express our results as the motion divided by the distance to the central star, which would be a constant for homologous expansion. The errors in the position for individual features expressed in relative motion were typically 0.02 mas yr⁻¹ arcsec⁻¹. For F658N this method gave for 20 objects a global average for the expansion rate of 0.23 ± 0.08 mas yr⁻¹ arcsec⁻¹ and for four F502N objects 0.33 ± 0.12 mas yr⁻¹ arcsec⁻¹, for a weighted average of 0.24 ± 0.11 mas yr⁻¹ arcsec⁻¹. These values are slightly larger than the results from simply comparing the scaled images; however, these are to be preferred because of their more quantitative derivation. In this case individual features are measured and an estimate of their accuracy is determined from the scatter of the results. In the scaling method described in Section 3.1 there is only the qualitative judgment of when the scaling is correct, with the uncertainty estimated by when the scaling would become unreasonable. There were no statistically significant differences in the expansion properties in the four quadrants sampled.

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We determined by comparing the P.A. of the measured motions with the P.A. from the central star that possible slight misalignment of the images in rotation did not strongly affect our results. We found that on the average the motion vectors were 5° smaller. This could be the result of a misalignment of the first and second epoch images of only 00116 and this would affect the derived expansion motions by only about 0.05%.

We also measured features within the nebula employing a different approach. The method was to create samples of 40 wide along P.A. values selected to lie along the major and minor axes of the nebula and intermediate angles displaced by 45°. Averages were taken perpendicular to the P.A. direction and profiles were created. Well-defined peaks not associated with knots were then measured in the first and second epoch images and in both filters. Fewer [O iii] features were measured because of the more amorphous appearance of the nebula in the F502N filter. The results are shown in Figure 3 and Table 5. The expansion rate was 0.30 ± 0.17 mas yr⁻¹ arcsec⁻¹ for F502N, and 0.20 ± 0.09 mas yr⁻¹ arcsec⁻¹ for F658N. Because the F502N features are broader and more difficult to measure, they were assigned half the weight of the F658N features in determining the average expansion rate for the nebular features of 0.22 ± 0.11 mas yr⁻¹ arcsec⁻¹.

The electron densities can in principle be determined from doublet line ratios with little dependence on the assumed electron temperature. The [S ii] 6717 and 6731 Å doublet ratio gives densities at the MIF, while the [Cl iii] 5517 and 5537 Å doublet largely arises in the He⁺ zone, with the results shown in Table 2. Garnett & Dinerstein (2001) used their higher resolution spectra to determine densities in the He⁺ zone from the [Ar iv] 4711 and 4739 Å ratio, after correcting the 4711 Å for the unresolved He i 4713 Å line by scaling from the He i 4471 Å line, finding densities of about 700 cm⁻³, which are in basic agreement with our [Cl iii] results for this same region.

The electron temperatures can be obtained from ratios of lines having very different excitation levels. The most widely used is the F(4363)/[F(5007) + F(4959)] of [O iii], which arises in the He⁺ zone. Since the 5007 Å line was saturated in the bright-ring portion of even our shortest exposure time spectra, we determined the F(5007)/F(4959) from the unsaturated central section of the spectra and found this to be 2.94. The value predicted using the transition probabilities presented in Osterbrock and Ferland (2006) is 2.9 and the observed value could be used to refine the determination of the transition probabilities. The results are also given in Table 2 and are in good agreement with those of earlier studies (Barker 1987; Guerrero et al. 1997; Garnett & Dinerstein 2001; Liu et al. 2004). The [N ii] F(5755)/[F(6548) + F(6583)] gives temperatures in the He^o zone which are similar to those in the He⁺ zone, which is not surprising since most of the heating arises from Lyman continuum photons and that radiation field has suffered little modification in these regions. In even finer sampling of our spectra we see the same rise in the [O iii] and [N ii] temperatures toward the central star that was found by Garnett & Dinerstein (2001) in [O iii], except that the local peak in temperature at about 10'' on both sides of the central star is better defined. This position corresponds to the inner boundary of a peak region of He ii 4686 Å emission, which suggests that there is a central lower density but higher temperature region along the polar axis of the nebula. Temperatures are determined in the MIF by the [F(4068) + F(4076)]/[F(6731) + F(6731)] ratio of [S ii] and the F(5577)/[F(6300) + F(6363)] lines of [O i]. The intrinsically weak 5577 Å line is absent in our middle region samples, thus precluding derivation of a temperature there and the F(4068) + F(4076) lines are contaminated by line of sight (LOS) higher ionization lines (in particular O ii recombination lines).

In our earlier paper (O'Dell et al. 2007b) that produced a three-dimensional model of the nebula, we presented the results of calculations with the Cloudy (Ferland et al. 1998) code using the appropriate stellar and nebular conditions. Excellent agreement was found between the observed and calculated properties for both the axes lying nearly in the plane of the sky. Less good agreement was found for the more poorly defined structure nearly along the LOS, where the predicted nebular density was too small. This discrepancy is exacerbated by our spectroscopic determination that the densities in the outer parts of these polar regions (as determined from the [S ii] doublet in the middle samples) are no lower than in the outer parts of the planar region. This leaves a quandary about the structure in the polar region, as both the arguments of O'Dell et al. (2007b) and Steffen et al. (2007) for the ionization of the outer halo call for this region being optically thin.

Arguments for the existence of homologous expansion in the Ring Nebula appear in both the radial velocity data (O'Dell et al. 2007b) and in our tangential velocity data. As noted earlier, this type of motion defies explanation by hydrodynamic models, but it cannot be ignored because it has been well established for radial velocities for about half a century (Wilson 1950; Weedman 1968).

In the case of the present study of tangential velocities, the average of the results from measuring motions of the knots and the nebula give a tangential expansion relation of V_tan= 0.23± 0.10 × ϕ mas yr⁻¹, where ϕ is the distance from the central star in the plane of the sky in seconds of arc. The relation of this result to the radial velocity expansion relation is determined by the distance to the nebula (D).

By comparing the ratio of radial velocities with respect to the systemic velocity (V_r) at various angular distances from the central star, O'Dell et al. (2007b) determined that V_r/ϕ  = 0.98 km s⁻¹ arcsec⁻¹, without giving an uncertainty. The relation of V_r to the spatial expansion velocity (V_s) will be determined by the geometry of the spheroid best describing the nebula. O'Dell et al. (2007b) argue that the structure perpendicular to the main ring of the nebula is a spheroid with a vertical length (H) 1.5 times the major axis of the elliptical main ring. In this case, the spatial expansion law would be V_s= 0.65 × ϕ_s km s⁻¹, where ϕ_s is the angle corresponding to the distance between a point in the nebula and the central star.

We can use our new results to more accurately determine the tilt of the plane of the elliptical main ring of the nebula with respect to the plane of the sky (θ). One knows from the nontilted form of the velocity ellipses of the high-resolution spectra made nearly across the minor axis (P.A. = 160°) that this axis lies in the plane of the sky, whereas those nearly along the major axis (P.A. = 70°) are tilted with a difference of velocity of 6 ± 2 km s⁻¹, with the east–northeast end approaching the observer. Assuming that the full major axis is 67'' and using the expansion rate of 0.23 ± 0.10 mas yr⁻¹ arcsec⁻¹ means that the tangential velocity is 25.6 km s⁻¹. The tangent of the tilt angle will be the ratio of the radial velocity difference (3 km s⁻¹) divided by the tangential velocity (25.6 km s⁻¹). This yields a tilt angle of 6.7°± 2° and decreases about linearly if the true distance is greater than the adopted distance of 700 pc. This is in excellent agreement with the derivation from only the radial velocities of 6.5°± 2°, the result presented in O'Dell et al. (2007b). This is small enough to simplify the calculation of the distance.

There are two complicating factors that need to be considered carefully when deriving a kinematic distance to the nebula. The first is that the radial velocities derived from spectroscopy are most sensitive to the expansion of those regions of the nebula where the expansion direction is closest to the LOS. In the case of the Ring Nebula, these are the polar regions of the nebula. The proper motions, on the other hand, will be more representative of the equatorial regions of the nebula, where the expansion direction is closest to the plane of the sky. In order to deal with this complication, it is necessary to adopt some model for the internal kinematics of the nebula. In this paper, we follow O'Dell et al. (2007b) in assuming a "Hubble flow" (all gas velocities strictly radial and linearly proportional to distance from the central star), with a nebular radius along the polar axis of 1.5 times that along the major equatorial axis at P.A. = 70°. Although this simple kinematic model seems to be broadly consistent with the observations, it is important to test it by searching for deviations from a pure Hubble flow, as emphasized by Steffen et al. (2009).

The second complicating factor is that the proper motions measure a pattern speed, whereas the spectroscopy measures the physical speed of the gas. These are not necessarily the same thing, as elucidated in detail by Mellema (2004). In Appendix A, we derive the correction factor, $\mathcal {R}$ (ratio of pattern speed to physical speed) as a function of two parameters: f₀, which is the degree of concentration of the ionized gas at its outer edge, and β_*, which is the rate of change of the stellar ionizing luminosity divided by the nebular expansion rate: $\mathcal {R} = 2 f_0 /(2 f_0 - 1 - \beta _*/3)$. The density profile found from fitting Cloudy models to nebular imaging and spectroscopy (O'Dell et al. 2007b) shows a roughly linear increase with radius, implying f₀ ≃ 1.3. The dynamic timescale (O'Dell et al. 2007b) and the stellar evolution timescale (Blöcker 1995) are roughly equal at 4–5000 years, implying (β_* ≃ −1). Hence, we find $\mathcal {R} \simeq 1.34$.

We can derive the distance to the Ring Nebula by assuming that the component of the spatial expansion (V_s) in the plane of the sky is equal to the corrected tangential component, the former being determined from the radial velocity study (O'Dell et al. 2007b) and the latter using the tangential angular motions determined in the present study. Equating these two components one has V_s × cos θ = 4.74 μ × D × $\mathcal {R}$⁻¹, where units of arcsec year⁻¹, km s⁻¹, and parsecs are used. Within the assumption of homologous expansion, the distance from the central star drops out. Employing the velocity expansion relation (V_s = 0.65×ϕ_s km s⁻¹) determined from the radial velocities, the expansion rate of 0.23 ± 0.10 mas yr⁻¹ determined from the observed tangential angular velocity, and the value $\mathcal {R}$ = 1.34 from Section 5.3, one finds D = 790⁺⁵⁰⁰₋₂₀₀ pc. This is larger, but within the uncertainties of the parallax determined value of 700±⁴⁵⁰₂₀₀ pc (Harris et al. 2007). Since both methods determine the distance from the reciprocal of the measured quantity, it is illuminating to compare the measured parallax 1.42 ± 0.55 mas with our predicted parallax 1.27 ± 0.73 mas. Taking the average of these parallaxes (1.35 ± 0.6), one determines a best recommended distance of 740⁺⁵⁰⁰₋₂₀₀ pc. This is only a small change from the parallax determined distance used previously, but it is determined from two independent methods and is the value to now be preferred.

The origin of the knots in the Ring, Helix, and other nearby PNe (O'Dell et al. 2002) remains highly uncertain and an important diagnostic observation is whether the knots share the general expansion characteristics of the host PN or if they trail behind. In a study of the radial velocities of the similarly nearly pole-on PN NGC 7293 (the Helix Nebula) Meaburn et al. (1998) argued that the knots were expanding, but at about one half the rate of expansion of the nebular material. This interpretation of the radial velocity data was disputed (O'Dell et al. 2004) when the knot velocities were compared with several sources of nebular velocities, the conclusion being that the expansion rates of the knots and the nebula are indistinguishable. That also appears to be the case for the Ring Nebula, since the plane of the sky expansion rate for the knots (0.24 ± 0.11 mas yr⁻¹ arcsec⁻¹) is not distinguishably different from that for the nebular features (0.22 ± 0.11 mas yr⁻¹ arcsec⁻¹). This means that either the knots in the Ring Nebula move in a very different way than those in the Helix Nebula or that the Helix Nebula results based on radial velocities (Meaburn et al. 1998) are incorrect.

The images of the Ring Nebula made in the H₂ 2.12μ line (Speck et al. 2003) indicate that this emission arises from the same ionization-bounded knots that produce the bright cusps seen in [N ii] and their dark central knots seen in [O iii]. The H₂ emission is probably excited by stellar radiation more energetic than 13.6 eV in the same manner as that in the Helix Nebula (O'Dell et al. 2007a; Henney et al. 2007). A lower spatial resolution Fabry Perot study (Hiriart 2004) at a resolution of 24 km s⁻¹ also noted that the 2.12μ emission arose in these features. Hiriart (2004) used his data to derive a model in which the 2.12μ emission comes from a cylindrical shell seen almost along its rotational axis, which is fully consistent with the triaxial spheroid model (O'Dell et al. 2007b) that has most of the material lying in an equatorial disk lying nearly in the plane of the sky. Using the H₂ velocities and distances from Hiriart's study, one finds a ratio of expansion velocity to angular separation of 1.1 km s⁻¹ arcsec⁻¹, which is within the determination errors of the value of 1.0 km s⁻¹ arcsec⁻¹ derived from the optical line study (O'Dell et al. 2007b; see Section 5.1), which had a velocity resolution of 7 km s⁻¹. This agreement in radial velocity adds credence to the interpretation that the knots are moving at the same velocity as the nebular material.

On a small-scale, many planetary nebulae show inhomogeneities in the form of dense molecular knots (O'Dell et al. 2002). In the case where the knot is well inside the global ionization front, there is expected to be a locally divergent flow of ionized gas away from the head of the knot, e.g., Bertoldi & McKee (1990). This contradicts the assumption that all gas flows are radially away from the central star, so the above analysis cannot be applied directly to this case. However, the ionization front conditions on the knot should be approximately D-critical (U − u = c_i, where c_i is the isothermal sound speed in the ionized gas). Therefore, one finds $\mathcal {R} = 1 + \mathcal {M}^{-1}$, where $\mathcal {M} = u/c_\mathrm{i}$ is the Mach number of the ionized gas at the surface of the knot on a radial line from the central star, as measured in the rest frame of the star. In order to apply this formula, it is necessary that both the proper motion observations (which measure U) and the spectroscopic observations (which measure u) are detecting the knots, rather than the diffuse nebular emission. This requires a very high angular resolution, which is not typically available from ground-based spectroscopy.

In Section 2.2, we established that the visual brightness (V= 15.65) of the central star was 0.1 mag brighter than assumed previously (O'Dell et al. 2007b), but that the extinction was less (c_Hβ = 0.13, A_V) than had been previously assumed (c_Hβ = 0.20, A_V = 0.43), so that this would not significantly change the previous result that the absolute luminosity of the central star is about 200 L_☉, (these new numbers argue for its being 5% fainter). The larger recommended distance (740 pc rather than 700 pc) argues that the star is an additional 11% brighter. This means that at present there are no good grounds to question our previous conclusions about the evolutionary state of the central star, that is, that it is a star of about 0.61–0.62 M_☉ about 7000 years beyond the end of the asymptotic giant branch (AGB) wind.

The ionizing luminosity of the central stars of planetary nebulae first rises with time as the post-AGB atmosphere contracts toward higher effective temperatures, and then falls swiftly when envelope nuclear burning ceases, followed by a much slower decline as the white dwarf interior cools (Blöcker 1995). The overall timescale for this sequence is set primarily by the core mass at the end of the AGB phase (higher mass cores tend to evolve more rapidly), but with a secondary dependence on the AGB mass-loss history. The expansion timescale of the nebula generally increases monotonically with the nebular age, so that the three phases in the luminosity evolution (rise, swift decline, slow decline) produce three phases in the nebular evolution: ionization (β_* > 0), recombination (β_* < −3), reionization (−3 < β_* < 0). These phases can be seen, for example, in the evolution of the mass of ionized gas in the models of Natta & Hollenbach (1998).

The detailed evolution of the nebula can be rather intricate, even when only one-dimensional models are considered (Perinotto et al. 2004), especially during the high-luminosity phase when the fast stellar wind from the central star is expected to be dynamically important in the inner regions of the nebula. Nonetheless, so long as the nebula is optically thick to ionizing radiation, the simple equation given above can be reliably applied in order to correct expansion parallaxes measured from the main nebula shell for cases where detailed tailored models, such as those given in Schönberner et al. (2005), are not available.⁵ The [$\mathrm{N\,\scriptstyle II}$] line is to be preferred over [$\mathrm{O\,\scriptstyle III}$] for both spectroscopic and proper-motion measurements, since it traces gas close to the ionization front, giving the most reliable estimates of u and U. Note that the model given here does not apply to density-bounded nebulae, where the ionizing luminosity is high enough to ionize all the circumstellar material. The lack of an ionization front means that other features, such as shocks (Mellema 2004), must be used to determine expansion parallaxes in such cases.

It is instructive to consider some particular limits of the general Equation (A6).

No stellar evolution. If the ionizing luminosity, Q_H, is constant, then β_* = 0 and we have $\mathcal {R} = 2 f_0 /(2 f_0 - 1)$. For a constant density nebula, f₀ = 1 and we obtain $\mathcal {R} = 2$, which is the well-known result for the evolution of a classical Strömgren sphere during the subsonic, D-type phase—the maximum ionized gas velocity is one half the expansion speed of the ionization front. On the other hand, for a thin, homogeneous ionized shell of thickness h ≪ R, one finds f₀ ≃ R/(3h) and $\mathcal {R} \simeq 1 + \frac{3}{2} h/R$, so that the correction factor is much reduced, tending toward unity as the shell thickness tends to zero.

Rising ionizing luminosity. If dQ_H/dt > 0, then β_* is also positive, which will always lead to larger values of $\mathcal {R}$ than in the constant luminosity case. The greatest increase is seen for a constant density nebula, whereas larger values of f₀ tend to dampen the effect. As β_* → 6f₀ − 3, then Equation (A6) suggests $\mathcal {R} \rightarrow \infty$. However, the equation is not valid in this limit, which corresponds to a highly supersonic weak R-type ionization front, since for such fronts the advective term would need to be retained in Equation (A2).

Falling ionizing luminosity. If dQ_H/dt < 0, then β_* is negative, which causes a reduction in the correction factor, $\mathcal {R}$, with respect to the constant luminosity case. Irrespective of the value of the density factor f₀, if β_* < −3, then $\mathcal {R} < 1$, so that the gas velocity is now greater than the expansion pattern speed. In this case, the nebula is bounded by a recombination front, instead of an ionization front, and the mass of ionized gas falls with time.