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Astron. Astrophys. 358, 521-534 (2000)

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3. Mean radial distribution of embedded OB stars

Using the kinematic information provided by the [FORMULA] line for the detected IRAS sources, we have derived their galactocentric distances assuming purely circular motion about the galactic center. For simplicity we used a linear rotation curve and the standard IAU constants, [FORMULA] kpc and [FORMULA] km [FORMULA]. For lines of sight toward the galactic center and anticenter the VLSRs of the sources are small and may have large fractional errors due to systematic or random peculiar motions. Thus, for the analysis here we have excluded all sources within 10o of the galactic center and within 5o of the anticenter. Therefore, a total number of 748 detected sources have been used for the present analysis (Table 1), of which 349 (47%) are in the northern Galaxy (I and II quadrants) and 399 (53%) are in the southern Galaxy (III and IV quadrant).

After finding the galactocentric distance R for each source, they were binned in annuli [FORMULA] wide. The total number of sources in each ring was divided by the ring sampled area to evaluate the azimuthally averaged number surface density of massive star formation regions as a function of galactocentric radius north, south, and for the complete dataset (Table 2; Fig. 3). The derivation depends only on the assumption of pure circular motion, and uses the kinematic galactocentric distance of each source, which can be derived one-to-one from the CS(2-1) velocity profiles.

[FIGURE] Fig. 3. The left column shows the number of detected sources divided by the sampled area, as a function of R, for the northern ([FORMULA]) Galaxy (top row), the southern ([FORMULA]) Galaxy (medium row), and the whole Galaxy (bottom row). The center column shows, in the same order, the azimuthally averaged face-on FIR galactic surface luminosity produced by embedded massive stars; the right column shows the azimuthally averaged FIR luminosity per embedded massive star forming region. The solid line corresponds to the complete dataset. The dashed line represents the dataset corrected for incompleteness of the sample, as described in Sect. 3.5


[TABLE]

Table 2. The embedded massive stars layer


3.1. Centroid and thickness of the OB star formation layer

To derive the azimuthally averaged centroid [FORMULA] and thickness [FORMULA] (HWHM) of the massive star formation layer we need to calculate the first and second order moments of the Z distribution for each of the galactocentric rings analyzed. Within the solar circle, however, for each source there are two possible kinematic heliocentric distances allowed by the rotation curve and, therefore, two possible values of Z. Following Paper I, we assume that the distribution [FORMULA] of sources is of gaussian form in Z,

[EQUATION]

where R is the galactocentric distance, [FORMULA] and [FORMULA] are fixed for each ring and A is a normalization constant. We then calculate the first and second order moments of the Z distribution in each ring, [FORMULA] and [FORMULA], by evaluating a statistical average, using weights that depend on the near or far distance from the plane, [FORMULA] and [FORMULA]:

[EQUATION]

and

[EQUATION]

where [FORMULA] is the number of sources in a ring at galactocentric distance R, [FORMULA], [FORMULA], and the normalization condition [FORMULA] is used.

The centroid [FORMULA] is equal to the first order moment [FORMULA], and the thickness [FORMULA] (HWHM) is given by

[EQUATION]

The weights [FORMULA] and [FORMULA] depend in turn of [FORMULA] and [FORMULA] at each ring, so the problem is cyclic. It can be solved by an iterative procedure (Casassus 1995), starting with reasonable values, like [FORMULA] and [FORMULA] pc, close to the average HWHM of the H2 layer within the solar circle.

While it is clear that each source is either at the near or at the far distance within the solar circle, the weights used in the statistical averages account for the probability of each source to be at the near or far distance by using the well known latitude effect , i. e., the farther away a source, the more likely to be close to the galactic plane. The weights are used only to derive large scale averages in a statistical sense, and not individually to derive a mean distance for each source. The results of the iterative calculation are displayed in Fig. 4 and listed in Table 2; empty spaces in the table correspond to those rings where the number of detected sources is less than 3, not enough to have a convergent iteration so as to derive values for [FORMULA] and [FORMULA].

[FIGURE] Fig. 4. The azimuthally averaged centroid [FORMULA] (top panel) and the thickness (HWHM) [FORMULA] of the massive star formation layer (bottom panel) are shown for the northern ([FORMULA]) Galaxy (left); the southern ([FORMULA]) Galaxy (center); and the complete Galaxy (right)

3.2. Azimuthally averaged FIR surface luminosity produced by embedded OB stars

The azimuthally averaged face-on FIR surface luminosity is evaluated by summing the luminosities of the sources in each ring and dividing the result by the sampled area. The average luminosity per source is the ratio of the summed luminosity in each ring to the number of sources considered. We estimate the FIR flux of a source as the sum over the four IRAS bands,

[EQUATION]

where [FORMULA] are the IRAS band flux densities, as listed in the IRAS PSC (Pérault 1987). The FIR luminosity for each source, outside the solar circle, is computed as [FORMULA], where D is its kinematic distance, obtained from its CS(2-1) line profile. We overcome the two-fold distance ambiguity, within the solar circle, by evaluating the total luminosity in each ring, [FORMULA], as a weighted sum. Like for the Z moments, in that sum we use the latitude effect to take into account the probability for each source to be at the near or far distance in the galactic disk, i.e.,

[EQUATION]

where [FORMULA] and [FORMULA] are the luminosities that the source would have at the near and far distances, respectively. The weights [FORMULA] and [FORMULA] are obtained using the values of [FORMULA] and [FORMULA] listed in Table 2. For the innermost ring, at [FORMULA] = 0.25, where no model is available, and at [FORMULA] = 0.35 in the north, we use the subcentral distances to estimate the luminosities. The FIR surface luminosity produced by regions of massive star formation and the average FIR luminosity per region, as a function of R, are shown in Table 3 and Fig. 3. The procedure used to remove the distance ambiguity, within the solar circle, is fully equivalent to the direct partitioning technique used to derive the H2 surface density in Paper I (see Eq. 4 and Fig. 15).


[TABLE]

Table 3. FIR surface luminosity


3.3. Errors derivation

The errors in the number surface density N consider only the counting statistics assuming a Poisson distribution. The uncertainties in the best values of the parameters [FORMULA] and [FORMULA] were estimated on the basis that the errors are dominated by the peculiar motions of the sources, which we assume to be of [FORMULA] km s-1 (VLSR). Using this value we have then calculated the uncertainties in the kinematic distances, and in the quantities which are derived from the kinematic distances. At [FORMULA], in the northern Galaxy, the average luminosity per source is dominated by one source, G70.293 (BNM), a very bright UC H II region in the Perseus spiral arm.

The uncertainties in the [FORMULA] values have also a contribution from the instrumental errors in the IRAS detectors. We used the errors in the 100 µm band as representative of the fractional flux uncertainty for each source (Casassus et al. 2000). When the flux value obtained by IRAS is only an upper limit, the error has been assumed to be of -100%. There is also some level of correlation in the position of the sources; as an example, for massive star forming GMCs in the IV galactic quadrant there are [FORMULA] embedded UC H II regions per GMC (Bronfman 1992). We have taken into account such correlation by computing for each galactocentric ring [FORMULA], the average number of sources that lie within a box of sides 2o by 2o by 20 km s-1 - a representative maximum value for a GMC - and multiplying the errors in [FORMULA] at each ring by [FORMULA].

3.4. Spiral structure and the axisymmetric analysis

What are the shortcomings in our analysis caused by deviations from axial symmetry in the galactic distribution of embedded OB stars? Images of external galaxies clearly show a high concentration of massive star formation in spiral arms, and the same should be expected for our own Galaxy. While the large scale distribution of OB stars in the Milky Way cannot be traced optically because of dust obscuration in the disk, H II regions produced by these stars can be observed in radio-recombination lines, like H 109[FORMULA], and have been used to study the spiral arm structure of our Galaxy (Georgelin & Georgelin 1976; Downes et al. 1980). The UC H II regions, which point at very young star forming regions, still embedded in their parental molecular clouds, could be even more confined to spiral arms than the extended H II regions observed in centimetric wavelengths.

Spiral arms within the solar circle appear to be more separated and less difficult to trace in the southern Galaxy than in its northern counterpart. These arms, as outlined by CO emission from molecular clouds, are tangent to the line of sight at longitudes [FORMULA] (Carina); [FORMULA] (Crux); [FORMULA] (Norma); and [FORMULA] (3-kpc expanding arm). The tangent longitudes are traced by discontinuities in (a) the galactic CO emission integrated in velocity and latitude I(l) (Paper I; Grabelsky et al. 1987) and in (b) the rotation curve derived from CO observations (Alvarez et al. 1990). The distribution of H II regions in the southern Galaxy, derived from an extensive survey of hydrogen recombination lines toward 5 GHz continuum sources (Caswell & Haynes 1987), is consistent also with the presence of these four spiral arm segments in the IV galactic quadrant.

To test our axisymmetric analysis we have derived the mean radial distribution of UC H II regions in the southern Galaxy in the framework of this generally accepted spiral arm model. Most GMCs identified through CO observations in the southern Galaxy are coincident, in space and velocity, with one or more radio H II regions from the Caswell & Haynes (1987) sample, as well as with several UC H II regions from the dataset we analyze here. We have used those radio H II regions from Caswell & Haynes that have well defined distances, derived through H2CO absorption lines, to remove the distance ambiguity for the associated GMCs and for their embedded UC H II regions (Bronfman 1992). These GMCs, together with the tangent points listed above, trace a spiral arm pattern in the longitude-velocity diagram (Fig. 5). The distance ambiguity for the remaining GMCs is removed by assigning each one to the closest spiral feature in the longitude-velocity diagram. For some GMC's in the far side of the Crux arm, close to the solar circle, we use their size-to-linewidth ratio to distinguish them from local clouds with small VLSRs.

[FIGURE] Fig. 5. The spiral arm pattern used for the analysis. The arms are labelled 1 (Carina), 2 (Crux), 3 (Norma), and 4 (3-kpc). The CO emission (Paper I; Grabelsky et al. 1987) integrated in latitude from [FORMULA] to [FORMULA] is represented in gray-scale. Lowest contour level is at 3 [FORMULA]. The scale in the inset figure is in kpc, and the galactic center is labelled G.C.

The mean radial distribution of FIR surface luminosity produced by embedded UC H II regions, obtained using the spiral model, is fairly similar to that obtained using the axisymmetric analysis (Fig. 6a). The average luminosity per source (Fig. 6b) shows a secondary peak, close to [FORMULA], caused by the far side of the Crux spiral arm cutting through the solar circle at [FORMULA]. We do not quote errors in this comparison because the spiral model, although well updated, involves much uncertainty and we are mainly interested in the general trend of the mean radial variations. It is apparent that the peaks are at the same positions and the general shapes are the same for both analyses. The axisymmetric analysis, however, seems to overestimate the mean FIR luminosities for radii other than the main peak locus.

[FIGURE] Fig. 6. a  FIR surface luminosity and b  average luminosity per source, as a function of R, for the southern Galaxy. The solid line shows the results obtained by removing the two-fold distance ambiguity, within the solar circle, using a spiral model. The dotted line shows the results obtained using the axisymmetric analysis

3.5. Completeness of the sample and contamination by low luminosity sources in the solar neighborhood

The question of completeness has been adressed in general by WC89, who show that UC H II regions are tightly confined in the FIR color-color plot, and can be easily distinguished from other entries in the IRAS PSC. Thus, the distinctive FIR colors of embedded massive stars can be used to count the number of such objects in the Galaxy. They conclude that all embedded O stars and some B stars will be detected by the color-color criterion applied to the IRAS PSC in the Galaxy. Contamination of the sample by cloud cores with lower mass stars has been analyzed by Ramesh & Sridharan (1997), who argue that the total number of potential UC H II regions in the WC89 sample is only about 25% of the sample. Our own criterion, based on the detection of the CS(2-1) line toward each of the UC H II region candidates (BNM), brings down the original WC89 sample by a factor of [FORMULA].

We analyze here the question of completeness of the sample as a function of distance; this is particularly necessary to understand the behaviour of the number density and mean luminosity of sources near the solar circle. The FIR luminosity versus distance for all sources in the sample analyzed is shown in Fig. 7. The fairly homogeneous distribution in distance of sources with FIR luminosities larger than [FORMULA] suggests that our sample is complete above such luminosity level. Closer than [FORMULA] kpc from the Sun the sample appears to be contaminated by lower luminosity sources. To test our results we have reanalyzed our data applying a lower FIR luminosity cutoff to the sample, both for the spiral and for the axisymmetric model. Wouterlout et al. (1995, their Fig. 20b) show that the WC89 FIR color criterion for UC H II regions tends to select point sources with [FORMULA]. Thus, even if the IRAS PSC were more sensitive, there would be little point in reducing the luminosity limit much under [FORMULA].

[FIGURE] Fig. 7. FIR luminosity versus distance for all the sources in the sample. Within the solar circle (left) the dots represent distances and luminosities obtained through the weighting method (See Sect. 3), while the circles represent subcentral sources, with better determined distances. Outside the solar circle (right) all distances are well defined

The CS(2-1) detection requirement appears not to introduce additional biases so we consider unnecessary to address separately the completeness of the CS survey. The observed antenna temperatures [FORMULA] within the solar circle, in particular for the subcentral sources, are not distance dependant. For sources outside the solar circle, the average [FORMULA] is 0.9 K for distances between 2 and 6 kpc, and 0.6 K beyond. The SEST beamwidth of 50" (FWHM) projects a linear size of 2.4 pc at 10 kpc, about the typical size of CS cores harbouring massive star formation (Bronfman 1992), so beam-dilution is not a stringent problem for the CS detections.

The mean radial distribution of the FIR surface luminosity does not change noticeably when a lower cutoff luminosity is applied to the sample. We have computed the azimuthally averaged FIR surface luminosity and the mean luminosity per source as a function of galactocentric radius, using the same spiral analysis as in Sect. 3.4 within the solar circle, and a lower cutoff luminosity [FORMULA]. The results are presented in Fig. 8, and the number of sources above the cutoff is listed, for each ring, in Table 1 (in parenthesis). At the cutoff value used, the three peaks in the average luminosity per source have the same magnitude, i.e [FORMULA]. These peaks are dominated by sources located in the far sides of the Norma arm ([FORMULA]), the Crux arm ([FORMULA]), and the Carina arm ([FORMULA]).

[FIGURE] Fig. 8. a  FIR surface luminosity and b  average luminosity per source, as a function of (R), in the southern Galaxy. The two-fold distance ambiguity, within the solar circle, has been removed using a spiral model. The results obtained using the complete dataset are shown with the solid line; results obtained when leaving out all sources with FIR luminosities lower than 8000 [FORMULA] are shown by the dotted line

While we cannot, for the time being, confirm that the average luminosity per source diminishes with galactocentric radius, it is tempting to suggest that UC H II regions appear to be very good tracers of spiral arm structure in the Galaxy, and that a full spiral analysis of their distribution in the Milky Way is mandatory. While in the present paper we perform a zero'th order axisymmetric comparison of the distribution of OB star formation regions and of molecular clouds, there is an evident necessity of resolving the two-fold distance ambiguity for all these regions in order to refine the present results, within the solar circle, and extend them toward a more exact analysis of the mean luminosity per source, which has a direct impact on the determination of the IMF for massive star forming regions.

The FIR surface luminosity derived from the axisymmetric analysis, our main result, remains also fairly unchanged (Fig. 3) when a lower luminosity cutoff is applied to the sample. The FIR surface luminosity at each radius is the product of the number of sources considered, per unit area, times their average luminosity. While the number surface density of sources decreases when a luminosity cutoff is applied, particularly for the solar neighborhood, the average luminosity per source increases, since we are not taking into account the less luminous sources for the average. Therefore the results we present next, based on our determination of the FIR surface luminosity produced by embedded massive stars and its comparison with the H2 surface density within the solar circle, appear to be not too affected by problems of completeness. As for the effects of spiral structure over the derived FIR surface density, they are small and would have presummably the same effects over the distribution of molecular hydrogen derived using an axisymmetric model.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000
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