2. FIR luminosity function of UCH II regions
The velocity information provided by the CS(2-1) observations from BNM allows, through the adoption of a rotation curve, to derive their galactocentric distances, and hence their heliocentric distances and luminosities. The galactic disk is assumed to be in circular motion, with kpc and km s-1. For the region of the Galaxy outside the solar circle (outer Galaxy) the procedure is a simple coordinate transformation from galactic longitude, latitude and velocity () to galactocentric radius, height over the plane and azimuth (). But such a transformation is bivalued for the region within the solar circle (inner Galaxy). There is a heliocentric distance ambiguity such that, unless a source lies just on the subcentral point (the tangent point to a galactocentric ring for a given longitude), there are two points along the line of sight, at the same distance on both sides of the subcentral point, that have the same line of sight velocity.
The method we used to resolve the distance ambiguity is described in BCMN. It is a statistical method that consists in weighting the near and far distances with a normal distribution in height over the galactic plane. Each source is assigned an effective luminosity, which is the weighted average of the near and far luminosities. The centroid and width of the vertical distribution, and , are determined through an iterative process for galactocentric bins wide. A consistency check for this method can be found below in this Section.
The kinematic distances are not reliable in the direction of the galactic centre, and also when the line of sight velocities are of the same order as the non-circular velocity components. Therefore, we excluded from the present analysis all sources within of the galactic centre, and within of the galactic anti-centre, as well as sources with . The resulting range in galactocentric radius, where the disk is properly sampled, excludes the solar circle. We restricted the analysis to sources with and .
The sources close to the subcentral points are an important consistency check of our method to resolve the distance ambiguity within the solar circle. The kinematic distance of a source at the subcentral point and in pure circular motion about the galactic centre is uniquely determined. The subcentral source sample was defined as the subset with line of sight velocities no more than 10 km s-1 different in absolute value from the terminal velocity (the maximum velocity expected for a given longitude in the case of circular rotation).
where are the IRAS band flux densities, as listed in the IRAS Point Source Catalog (1985). In order to test this approximation we compared with the total fluxes reported for 53 UCH II regions by Wood & Churchwell (1989b, WC89b, their Tables 17 and 18). Fig. 1a shows the ratio of the fluxes (obtained by Eq. 1) to the total fluxes of WC89 (which are integrated up to 100m), as a function of . Eq. 1 overestimates the WC89b fluxes by about a constant 20%, but WC89b did not include a correction for the 100m - 1 mm flux, which they estimate could be as high as 50%. We thus expect Eq. 1 to be a good estimate of the total luminosity of IRAS/CS sources within 30% (50% - 20%).
Since we will average the luminosity function of UCH II regions over large areas of the galactic disk, it is important to estimate the minimum luminosity above which the disk is properly sampled. The lowest flux in the sample, W m-2, corresponds to at a distance of 15 kpc. Fig. 1b shows a histogram of the total number of sources in our sample as a function of (without the velocity filter ). The triangles show the fraction of sources with only upper limits in the 100m band (right hand scale). Could a significant number of sources be missed by IRAS near the minimum detected flux? A lower sensitivity would be hinted at by an increased fraction of upper limits in the reported IRAS fluxes, which is not the case. However, the higher far-IR background towards the central regions of the Galaxy results in a completeness limit of at 8.5 kpc, within , . But over a broader longitude range, at 8.5 kpc, within , . As the luminosity functions of the subcentral sources (which are all within 8.5 kpc of the Sun) is in good agreement with that of the whole inner Galaxy (see below), we take as the completeness limit of the IRAS/CS sample 1. It should also be reminded the WC89a colour criterion selects UCH II regions candidates, which require spectral types earlier than B3, and luminosities greater than . Wouterloot et al. (1995, their Fig. 20b) show that the WC89a colour criterion for UCH II regions also selects point sources with - thus even if the IRAS PSC were more sensitive, there would be little point in reducing the luminosity limit much under .
The LF of the whole sample of IRAS/CS sources, our main observational result, appears to be significantly different inside and outside the solar circle; in Fig. 2 we distinguish between and . The LFs cover a very wide range in luminosity, over three orders of magnitude, which allows using logarithmic luminosity bins corresponding to a factor of 300%. Within the solar circle the LF obtained using the effective luminosities is confirmed to be a good estimate of the actual LF through its close similarity with the LF of the sources near the subcentral points, shown in dotted line 2. For comparison, placing all the sources at the `near' or `far' kinematic distance changes the peak of the LF as a function of logarithmic luminosity from 4.25 to 5.75, while the LF obtained using the effective luminosities peaks at 5.25. The good match with the subcentral source sample LF lends strength to a comparison of the luminosity functions between the outer and inner Galaxy, based on the effective luminosities. We will refer to the luminosity functions for the whole inner and outer Galaxy by and .
The dominant source of uncertainty in the LF is shot noise. The errors in the galactic disk surface FIR luminosity amount to about 10% upwards, 20% downwards (Fig. 3 in BCMN). The fractional error on the FIR surface luminosity represents the typical fractional error on the luminosity of one source. These errors stem from the IRAS 100m band flux uncertainty, coupled with the kinematic distance uncertainty due to non-circular motions of about 5 km s-1. Adding in quadrature the 30% uncertainty related to the use of Eq. 1, we have an average error on the luminosity of a source of at most 36%. Compared to the 300% width of the luminosity bins, a 36% uncertainty is negligible, apart from a slight smoothing effect without practical consequence.
The differences in the LFs inside and outside the solar circle are statistically significant. As a statistic for the difference between the inner and outer LFs, we used a test which has the following expression in this context,
where we sum over the bins above the luminosity limit of . The result is , or that and are different at a significance level of 95.6% (with a distribution for 5 degrees of freedom, corresponding to the number of bins with non-zero counts above the luminosity limit). For comparison, the same test applied to the northern and southern 3 LFs inside the solar circle gives a probability of 45% that the distributions are different, so they are comparable relative to the differences between the LFs inside and outside the solar circle. Another application of this statistical test gives that and the subcentral source LF are the same at a confidence level of 87%.
We emphasize the presence of a peak in the LF of IRAS/CS sources, well above the completeness limit. The strongest evidence in that sense can be found in the LFs for the outer Galaxy and for the subcentral sources, where the completeness limits are lowest. The shape for the LF we report is quite different from a power law functional form (e.g. as used in CT96).
© European Southern Observatory (ESO) 2000
Online publication: June 8, 2000