6 Luminosity-volume relation

Figure 10 shows the relation between the logarithmic HH$\alpha$ luminosity of a given HII region, log L (with L in erg s^-1), and the logarithm of the cube of the radius log r³ (r in arcsec). If this relation were to maintain a constant slope, this would mean that the product of the electron density, the proton density and the filling factor (fraction of the volume filled with ionized medium, Osterbrock 1974) were constant, independent of the luminosity of the HII regions. If regions of high luminosity were density bounded it would yield an increasing slope with increasing volume, i.e. the relation should show a concave upward curve, corresponding to an increase in the average filling factor. Although the scatter in Fig. 10 is large, there is an upward curvature at high luminosity, which is in agreement with the general scenario in which the most luminous HII regions in spirals (and in all probability in irregulars, see Fuentes 1997) are density bounded (Beckman et al. 1999; Rozas et al. 1996a). The observed distribution is discretized along the abscissa in Fig. 10 because r is calculated by counting the number of pixels within a region, dividing by $\pi$ and taking the square root, which leads to an obvious systematic scatter increasing fractionally to lower luminosities i.e. to smaller radii. There are further factors leading to uncertainty in the observed values in Fig. 10. These are: (a) faint regions are difficult to detect against the true diffuse background of HH$\alpha$ within the galaxy; this is especially so within the arms and the bar, though less of a problem in the interarm disc. (b) Overlap may occur especially in the arms and the bar, which may cause significant underestimation of the volume. In Fig. 10, we have plotted HII regions of the bar and of the disc, using different symbols. There is no clear difference between the properties of the two sets of regions characterized by this plot.

  

Figure 10: Relation between the logarithm of the cube of the radius of an HII region (r in arcsec) and the logarithm of its luminosity (in erg s-1). Fractional errors grow rapidly for log r3 < -0.5 due to the angular resolution limit in the image