EXPANSION PARALLAX OF THE PLANETARY NEBULA IC 418

The large uncertainties affecting distance estimates to planetary nebulae (PNe) continue to represent an important obstacle to our understanding of these objects. While the total number of PNe in the Milky Way has been estimated to be somewhere between 5000 and 25,000 (Peimbert 1990; Zijlstra & Pottasch 1991; Acker et al. 1992), less than 50 have distances that have been measured individually with reasonable accuracy. About one-third of these individual estimates are direct trigonometric parallax measurements (Harris et al. 2007), while the rest is based on a disparate set of indirect methods (e.g., spectroscopic parallax of the central star or of a resolved companion, cluster membership, reddening, or angular expansion). In addition, statistical methods, calibrated using PNe with individually measured distances, have been developed. Such a frequently used statistical technique (called the Shklovsky method) is based on the assumption that the mass of ionized gas is the same for all PNe. This rather crude assumption has been refined by Daub (1982) and Cahn et al. (1992) who distinguish between density-bounded PNe (for which the Shklovsky constant mass hypothesis is assumed to hold) and radiation-bounded PNe (where the mass of ionized gas is taken to depend on the surface brightness of the nebula). For a recent review of various methods of distance determinations to PNe, see Phillips (2002).

A potentially reliable direct geometric method for estimating the distance of an individual PN is the so-called expansion parallax technique. In this method, the angular expansion of the source on the plane of the sky is measured, and compared to the expansion along the line of sight determined from optical spectroscopy. Traditionally, the angular expansion has been obtained from multi-epoch radio interferometric data gathered over a period of several years (Masson 1986), but measurements at optical wavelengths with the Hubble Space Telescope (HST) have also been used (Reed et al. 1999; Palen et al. 2002). This technique has been applied successfully to several PNe (Masson 1986, 1989; Gómez et al. 1993; Hajian et al. 1993, 1995; Kawamura & Masson 1996; Hajian & Terzian 1996; Christianto & Seaquist 1998; Guzmán et al. 2006), and can provide distances accurate to about 20%. This is comparable to the accuracy obtained from existing trigonometric parallax measurements (Harris et al. 2007), and significantly better than the accuracy of statistical methods (e.g., Stanghellini et al. 2008). In this paper, we will apply the expansion parallax technique to the well studied PN IC 418.

The morphology of IC 418 (G215.2 − 24.2, PK 215 − 24.1, The Spirograph Nebula) is fairly simple: both at optical and radio wavelengths, it has an elliptical ring shape, with a major axis of 14'' and a minor axis of 10''. It is surrounded by a low-level ionized halo, which is itself enshrouded in a neutral envelope with an angular size of about 2' (Taylor & Pottasch 1987; Taylor et al. 1989). Widely discrepant estimates of the distance to IC 418 have been obtained using different statistical methods. To our knowledge, the shortest distance ever proposed is 360 pc (Acker 1978), whereas the largest one is 5.74 kpc (Phillips & Pottasch 1984). The Shklovsky method mentioned earlier gives a value of 1.9 kpc (Cahn & Kaler 1971). In recent years, the most popular value appears to have been 1 kpc (e.g., Meixner et al. 1996; Pottasch et al. 2004), although the reason for this is not entirely clear.

The data (Table 1) were collected with the Very Large Array (VLA) of the NRAO⁴ at 6 cm (5 GHz) in its second most extended (B) configuration on 1986 June 28 (1986.49) and 2007 November 6 (2007.85). This provides a time separation of 21.36 yr. The data were edited and calibrated using the Astronomical Image Processing System (AIPS) following standard procedures (see Table 1). The longest baselines were tapered down to increase the sensitivity, and self-calibration (in phase only) was applied. The resulting images are shown in Figure 1.

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Table 1. List of Observations

Project	Epoch	Phase Calibrator	Bootstrapped Flux Density (Jy)	Restored Beam
AP116	1986 Jun 28 (1986.49)	0605-085	2.347 ± 0.008	380 × 308; P.A.=0°
AL716	2007 Nov 06 (2007.85)	0516-160	0.425 ± 0.005	380 × 308; P.A.=0°

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The difference image between the two epochs (new minus old) was produced following Guzmán et al. (2006). It is shown in the bottom left panel of Figure 1, and exhibits the typical concentric negative/positive pattern expected when expansion is present. It is possible to estimate the expansion rate of the nebula by comparing the difference image of the data with a model. That model can be built from the data of either epochs, but the best results are obtained using the data set with the highest quality (corresponding to the second epoch in the present case). To generate the model difference, we image the data twice. The first imaging was made using a fixed pixel size of 01, whereas the second was made using pixels of (1 + )× 01, with ⩾ 0, but ≪ 1. We then performed pixel-to-pixel subtraction between those two images. The physical size of the source, of course, does not depend on the chosen size of the pixel. But since the pixels are larger in the second case, the source occupies a smaller number of them. Therefore, the pixel-to-pixel subtraction is equivalent to subtracting from the real image of the source, a self-similarly shrunken (i.e., a demagnified) version of itself. To identify the best model, we repeated the procedure described above for a set of values of . For each value, we compared the model difference with the real difference between the second and the first epoch. The best model clearly corresponds to the situation where the two differences are as similar to each other as possible. To obtain a quantitative measure of when that happens, we subtracted the two difference images, and calculated the rms of the resulting image. A plot of that rms as a function of (Figure 2) shows that the minimum of the rms corresponds to a well defined value of . That value was calculated using a fit to the data points with a quadratic form (blue curve in Figure 2). This yields a value of = 0.018 ± 0.005.

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The angular expansion rate can be determined from the value of using

$$\begin{eqnarray*} &&\dot {\theta } = {{\theta \epsilon } \over {\Delta t}}\. \end{eqnarray*}$$

The radius of maximum emission, θ, is estimated from the image at the second epoch to be 6700 ± 0006, so $\dot{\theta } = 5.8 \pm 1.5$ mas yr⁻¹. Since the radio emission considered here traces the ionized region of IC 418, $\dot{\theta }$ corresponds to the expansion rate of the ionization front.

To deduce the distance from the angular expansion rate calculated above, one must know the physical velocity v_exp at which the ionization front is expanding (e.g., Guzmán et al. 2006):

$$\begin{eqnarray*} &&\bigg [\frac{D}{\rm pc}\bigg ] = 211 \bigg [\frac{v_{\rm exp}}{\rm km\;s^{-1}}\bigg ] \bigg [ \frac{\dot{\theta }}{\rm mas\;yr^{-1}} \bigg ]^{-1}. \end{eqnarray*}$$

Traditionally, v_exp has been estimated using high spectral resolution observations of some emission lines and assuming a relation between the shape of the line profiles and the movement of the emitting gas. In the present case, we use the high-resolution spectra of Hβ, [N ii], and [O iii] lines published by Gesicki et al. (1996). The Hβ line is broad (FWHM = 18 km s⁻¹, dominated by the thermal contribution), while the [O iii] line is narrow and centered on the systemic velocity. The [N ii] line, on the other hand, is double-peaked with each peak at ± 20 km s⁻¹ around the systemic velocity. Using a detailed three-dimensional photoionization model of the nebula, Morisset & Georgiev (2009) reproduced these three profiles using a V ∝ R⁴ expansion law. According to this model, the expansion velocity near the outer edge of the nebula (where we detect expansion at radio wavelengths) is 30 km s⁻¹. This result is in good agreement with the radiation–hydrodynamic models of Schönberner et al. (2005) and Villaver et al. (2002). With this value of v_exp = 30 km s⁻¹, we obtain a distance to IC 418 of 1.1 ± 0.3 kpc.

There is a potential systematic effect discussed in detail by Mellema (2004) that ought to be taken into account. While the angular expansion measured using the VLA data corresponds to the progression of the ionization front (and is, therefore, a pattern velocity), the Doppler width deduced from spectral lines traces the expansion velocity of the material itself. For conditions appropriate for PNe, Mellema (2004) showed that the ionization front tends to expand somewhat faster than the material. Using their Figure 5, we estimate that the expansion velocity of the ionization front in IC 418 is 1.2 ± 0.1 faster than the expansion velocity of the gas. Taking this effect into account leads to a distance to IC 418 of 1.3 ±  0.4 kpc.

These two estimates of the distance are well within the broad range (0.36–5.74 kpc) of values obtained from statistical methods (see Section 1). They are also in very good agreement with the value of 1 kpc that has often been used recently, and with the value found by Morisset & Georgiev (2009): d = 1.26 ± 0.2 kpc.⁵

Two distance-independent parameters can be derived from our observations. First, the dynamical age of the nebula can be calculated from the angular size and angular expansion rate: $\tau _{\rm dyn} = \theta /\dot{\theta }$. We obtain τ_dyn ∼ 1200 yr. Second, the emission measure of the ionized region can be obtained from the observed parameters of the radio emission (we used the image at the second epoch, which is of better quality; the total 6 cm flux is 1.44 ± 0.01 Jy and the average deconvolved angular size of the emission is 134 ± 001). This yields an emission measure EM = (5.05 ± 0.01) × 10⁻⁶ cm⁻⁶ pc. Finally, the electron density and the mass of ionized gas can be calculated (also from the radio flux). We get $n_e = (6.2 \pm 1.7) ({d \over 1.1\rm \; kpc})^{-0.5} \times 10^3$ cm⁻³, and $M_i = (8.7 \pm 2.4) ({d \over 1.1 \rm \; kpc})^{2.5} \times 10^{-2}$ M_☉. This is in reasonable agreement with the values found by Morisset & Georgiev (2009) from their detailed modeling (n_e = 9 × 10³ cm⁻³ and M_i = 6 × 10⁻² M_☉).

In this paper, we presented observations of the 6 cm radio continuum emission from the well studied PN IC 418 obtained at two epochs separated by more than 20 years. These data allowed us to detect the angular expansion of the nebula, and to estimate its distance. Depending on the assumption made on the relative velocity of the matter and of the ionization front, we obtain a distance of 1.1 ± 0.3 kpc, or 1.3 ± 0.4 kpc.

L.G., Y.G., and L.L. acknowledge the support of DGAPA, UNAM, and CONACYT (Mexico). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.