3 Imaging

3.1 Observations

We obtained direct images of PNG 135.9+55.9  in narrow-band filters from two sites (Table 4). Images were obtained in the emission line of H$\alpha$ and a nearby continuum bandpass with the 2.1 m telescope in SPM. The Mexman filter wheel was used for observations. The seeing was modest, with stellar images having FWHM of 1 $.\!\!^{\prime\prime}$5 or worse. Images were also obtained using ALFOSC at the 2.6 m Nordic Optical telescope (NOT) on La Palma, Canary Islands, Spain. A narrow filter centred at H$\alpha$ was used, but the continuum was measured using a Stromgren y filter. The NOT images benefited from better seeing, $\approx$ $1\hbox{$.\!\!^{\prime\prime}$ }2$, on the first night of this run. The spectrophotometric standard stars HZ44 and Feige 67 were observed along with the object on the second night for calibration purposes. During both imaging runs, sequences of at least 3 images in each filter were obtained, with one long exposure usually not exceeding 20 min and two shorter ones. This tactic allowed us to keep cosmic ray rate within reasonable limits in order to later combine the images for improved statistics and to permit the removal of cosmic rays.

The images were processed using IRAF. The steps included bias substraction, flat field correction, image combination, and apperture photometry. The STSDAS package within IRAF was used for the deconvolution of the images.

The primary uses of the H$\alpha$ images were to determine the angular size of the nebula and to derive the density distribution of the gas. Two selected NOT images from the first night, where the seeing was better and the focus was good were combined to produce the image presented in Fig. 2. The nebula appears to be slightly elliptical in shape. No substructure is evident in the main body of the nebula. Its size comfortably exceeds the PSF of stellar objects, so the instrumental profile should not affect its shape. Nonetheless, we conducted series of deconvolution experiments on the nebular images using the Lucy-Richardson method (Lucy 1974; Richardson 1972) that showed that the size and shape of the nebula were unchanged after deconvolution. This is our strongest evidence that the central star does not emit significantly in H$\alpha$. For comparison the image of the central star obtained in the Stromgren y filter is presented in the bottom panel of Fig. 2 with the H$\alpha$ contours of the nebula superposed. We used the SPM observations, where the continuum filter was much closer in wavelength to the line filter, to substract the image of central stellar object from that of the nebula. Figure 3 presents the spatial profiles of the nebular emission in H$\alpha$ from the NOT image without subtracting the central star and from the SPM image after substraction of central star. Except for a slight depression around the peak, the shape of the nebula in both observations is very similar.

Figure 2: The upper panel presents the image of PNG 135.9+55.9 in the light of H$\alpha$, while the lower panel presents the Stromgren y image of the central star with the H$\alpha$ contours of the nebula superposed. The continuum and H$\alpha$ images were aligned using field stars. Both images were obtained with the NOT.

The NOT observations in the Stromgren y filter were used to estimate the magnitude of the central star. We inferred $m_{5556}=17.9\pm0.15$ mag, corresponding to a flux of F₅₅₅₆ =  $2.52\times 10^{-16}~{\rm erg~cm}^{-2}~{\rm s}^{-1}~{\rm\AA}^{-1}$ in a good agreement with the flux determined from spectroscopy. This continuum measurement should be reliable since there are almost no lines in the spectrum of this planetary nebula, the contribution of He II  $\lambda5413$ Å being negligible.

3.2 The nebular angular diameter and density distribution

The angular diameter of the planetary nebula is an important parameter since, in combination with the H$\beta$ flux, it allows a determination of the electron density and the nebular mass. We followed the recipe of Bedding & Zijlstra (1994) to assure that our measurements correspond to common standards. (For cautionary remarks, see van Hoof 2000.) There are a variety of definitions of the diameter of a PN in the literature: the FWHM, the surface brigthness contour at 10% of the maximum surface brightness, and the outermost contour at which emission is found. Since PNG 135.9+55.9 appears to have a slightly elliptical shape in the H$\alpha$ images, we have measured the diameter of the nebula along two axes. We find that the diameter of the nebula is 11 $.\!\!^{\prime\prime}$4$\times$9 $.\!\!^{\prime\prime}$8 if we use the outermost contour, 6 $.\!\!^{\prime\prime}$3$\times$5 $.\!\!^{\prime\prime}$2 if we use the 10% surface brightness contour, and 3 $.\!\!^{\prime\prime}$5$\times$2 $.\!\!^{\prime\prime}$95 if we use the FWHM.

Figure 4 shows the density and flux distributions adopted in our models. The symbols correspond to the observed H$\alpha$ emission profiles across the major and minor axes obtained from the SPM images (shown in Fig. 3). The curve corresponds to a spherical model with a density distribution described by $n = n_{\rm c} \exp -(\theta/2.8)^{2}$, where $\theta$ is the angular distance to the center in seconds of arc. In this model, the emissivity of H$\alpha$ is taken to be simply proportional to n²and is computed for a uniform temperature. Such a simple representation accounts very satisfactorily for the observed H$\alpha$ emission profile.

Figure 3: This figure presents the spatial profiles of the nebula from the H$\alpha$ images and of the central object from the 6650 Å continuum image. The filled and open squares represent cuts along major and minor axes of the nebula from the SPM images after subtraction of the continuum. Open triangles represent the spatial profile along the major axis of the H$\alpha$ image obtained at the NOT without continuum subtraction. The dotted line is the profile of the central star (SPM). The fluxes are normalized to the maximum intensity of the SPM continuum-subtracted image, except for the central star, which is normalized to its maximum intensity.

Figure 4: The density distribution used for the model calculations is presented in the lower panel. The predicted flux distribution from our models is drawn with a solid line in the upper panel, supposing that the emissivity is proportional to n2 and computed at a constant temperature. The open and filled squares denote the spatial distributions of the nebular emission along the major and minor axes, respectively, obtained from the SPM observations, as shown in Fig. 3. The abscissa denotes relative angular size normalized to the maximum, azimuthally-averaged radius of 5 $.\!\!^{\prime\prime}$6.

In the analysis of the photoionization models that follows, we adopt a nebular radius of 5 $.\!\!^{\prime\prime}$6, i.e., similar to the diameter of the outermost contour.