5 Physical properties

The fact that these images are absolutely calibrated allows physical properties to be derived from these observations. The total nebular fluxes in H$\alpha$, [N II], and [O III] are obtained by integrating over the nebular image for each of these filters. The corresponding logarithmic extinction values listed in Table 2 are taken from Cahn et al. 1992).

5.1 Nebular masses

The total flux $F_{0}(\rm H\alpha)$ emitted by the H$\alpha$ line from a uniform density sphere of radius R is given by

$$F_{0}({\rm H}\alpha) = \frac{4\pi j_\nu}{4\pi D^2} \left( \frac{4\pi R^3}{3} \right)$$ (3)

where $j_\nu$ is the line emission coefficient, and D is the distance to the nebula. For a recombination line, $j_{\rm H\alpha}$ is often expressed in terms of the effective recombination coefficient

$$\alpha^{\rm eff}=\frac{4 \pi j}{n_{\rm e} n_{\rm p} h\nu}$$ (4)

where $n_{\rm e}$ and $n_{\rm p}$ are the electron and proton densities respectively. The flux in the recombination line can then be written as

 

$$F_{0}({\rm H}\alpha) = \left( \frac{R^3}{3D^2} \right) h\nu... ...}\alpha} n_{\rm e} n_{\rm p} \alpha^{\rm eff}_{{\rm H}\alpha}.$$ (5)

In practice, a filling factor $\epsilon$ is often used to take into account the actually observed nebular structures which are far from homogeneous (see morphologies displayed in a recent report by Hua et al. 1998). Its value lies between 0.3 and 0.7. The ionized mass of the nebula is then given by

 

$$M_i = \frac{4\pi}{3} n_{\rm p} \mu m_{\rm H} \epsilon R^3$$ (6)

where $\mu$ is the mean atomic mass per H atom. Combining Eqs. (5) and (6), the ionized nebular mass can be expressed in terms of the angular radius ($\theta$) and H$_\alpha$ flux as

 

$$M_i=\frac{4\pi\mu {\rm m_H}}{\sqrt{3h\nu_{\rm H\alpha} x_{\r... ... \epsilon^{1/2} \theta^{3/2} D^{5/2} F_{0}({\rm H}\alpha)^{1/2}$$ (7)

where $x_{\rm e}=n_{\rm e}/n_{\rm p} \sim 1.16$. Since both $\theta$ and $F_{0}(\rm H\alpha)$ (i.e. our measured $F({\rm H}\alpha)$ fluxes corrected for extinction using radio - or optical when radio data are missing - $c_{\rm H\beta}$ values quoted in Table 2) are measured in our images, the ionized masses for the nebulae can be obtained as a function of their distances. The effective recombination coefficients for the recombination lines of H are tabulated in Hummer & Storey (1987). For example, $\alpha^{\rm eff}_{{\rm H}\alpha}$ at $T_{\rm e}=10^4$ K and $n_{\rm e}=10^4$ cm^-3 has a value of 8.65 10^-14 cm³ s^-1, and Eq. (7) can be written as

$$M_i=0.032 (\epsilon/0.6)^{1/2} \theta^{3/2} D^{5/2} F_{0}({\rm H}\alpha)^{1/2} \ M_{\odot}$$ (8)

with $\theta$ in arcmin, D in kpc, and $F_{0}({\rm H}\alpha)=F({\rm H}\alpha)\times 10^{0.693\times c_{{\rm H}\beta}}$ in 10^-12 erg cm^-2 s^-1 units. The derived values of M_i (assuming $\epsilon=0.6$) are given in Table 2. We adopted the major axis values for elongated PNe (A 13, A 36). Given the uncertainties in the distances, the derived ionized masses (mainly less than 1 $M_\odot$) for the other Abell PNe are not drastically different from most evolved PNe.