3 Analysis of data and discussion

We discuss the unidentified sources for which definite spectral type assignments have been made by DJM under the categories i) sources for which quality-3 flux density data are available at only 12 ${\mu}$m, ii) sources for which quality-3 flux density data are available at both 12 and 25 ${\mu}$m, and iii) sources for which quality-3 flux density data are available at 12, 25, and 60 ${\mu}$m.

3.1 B$_{\rm j}$-[12] colours of unidentified IRAS sources with quality-3 flux density data at only 12 ${\mu}$m

KVKI has carried out a search of the Guide Star Catalog (GSC Version 1) to obtain $B_{\rm j}$ magnitudes for the unidentified IRAS sources having quality-3 flux density data at only 12 ${\mu}$m (flux density upper limits in the other three bands) and having M-type spectral assignments by DJM. $B_{\rm j}$ is the magnitude determined from the UK-SERC Schmidt IIIa-J plates behind a GG 395 filter; the plates are primarily of fields south of the celestial equator. This subset of the unidentified IRAS sources is very likely to be characterised by extremely thin dust shells as their emission at 25 ${\mu}$m and 60 ${\mu}$m is below the IRAS detection threshold.

This search for optical counterparts has been restricted to the sources (numbering about 1075) lying in the 1950 R.A. range $5^{\rm h}$$18^{\rm m}$ to $9^{\rm h}$$30^{\rm m}$ and resulted in 320 classified IRAS sources meeting the above flux criterion for which $B_{\rm j}$ magnitudes are available. The star (when it is the only one) lying within 15$^{\prime\prime}$ of the IRAS source position has been taken as the optical counterpart of the source.

  

Figure 2: $B_{\rm j}$-[12] colour vs. spectral type of unidentified IRAS sources with best-quality flux density data at only 12 ${\mu}$m (which could be associated with optical counterparts) on the SERC plates by searching the Guide Star Catalog (Version 1). The optical object (when it was the only one) on the SERC plates within a circle of radius 15$^{\prime\prime}$ from the IRAS source position was considered as its optical counterpart. The $B_{\rm j}$ magnitudes are from GSC Version 1

  

Figure 3: Similar data as in Fig. 2 for BSC M giants. However, the BSC giants have IRAS quality-3 flux density data in the three bands 12, 25, and 60 ${\mu}$m unlike the IRAS unidentified sources with which they are being compared

Figure 2 presents $B_{\rm j}$-[12] colours as a function of spectral type for this group, and Fig. 3 presents similar data for southern BSC M giants. All these stars are of late spectral type, and it is well known that many late M stars are variable in light and spectrum and thus spectral types and the optical and infrared magnitudes will depend on the phase at which they are observed. We point out that no IRAS variability data are available to study the variability of these sources as a function of spectral type since they are detected with quality-3 flux densities at only $12~{\mu}$m and the requirement for a source to be listed as variable in the IRAS PSC is that the changes in the flux density at 12 ${\mu}$m and 25 ${\mu}$m be correlated. However, IRAS variability data are available (from IRAS PSC) for the BSC stars with which the colours of the IRAS unidentified sources have been compared. The incidence of variability of the BSC stars is quite low and does not show any definite dependence on spectral type. It is necessary to stress that the spectral type assignments of the sources discussed in this study are based on observations at epochs different from those for which the optical (from Sky Surveys) and the infrared magnitudes (from IRAS observations of 1983) are available. The differences in the epochs are expected to result in an enhancement of the scatter in the estimated colours (due to variability alone) as a function of spectral type. An additional source of scatter in the colours is due to differing amounts of interstellar extinction at $B_{\rm j}$ since interstellar extinction is near zero beyond 10 ${\mu}$m. The inclined lower bound to the points in these figures is presumably the locus of unreddened M giants with the thinnest CSE. The data on the mean $B_{\rm j}$-[12] colours as a function of spectral type for the unidentified IRAS sources and BSC M giants are presented in Table 2. The entries are as follow: Column (i) is the spectral subtype of the sources and in columns (ii) - (iv) and in (v) - (vii) are the number of sources of a particular spectral type, their mean $B_{\rm j}{-}[12]$ colour, and root mean square (rms) deviation in the value of the mean colour for the BSC stars and the unidentified IRAS sources, respectively. In Fig. 4 the mean values are given, and there is a noteworthy difference in the mean $B_{\rm j}$-[12] colours in the region of spectral type overlap albeit there are large statistical errors on the mean colours of the two groups. Kolmogorov-Smirnov (K-S) and Wilcoxon two-sided tests show that the two samples are not from the same population at the 95$\%$ confidence level. The difference is, no doubt, due to higher values of extinction toward the more distant, unidentified IRAS sources than toward the more local BSC M giants. Cohen et al. (1987) in their Fig. 4 presented V-[12] vs. spectral type for the BSC giants, and the trend for the M stars is similar to ours and is what is expected given the difference in magnitude systems.

  

Table 2: Mean $B_{\rm j}$-[12] colours of M-type BSC stars and IRAS unidentified sources

  

Figure 4: Mean $B_{\rm j}$-[12] colours of IRAS unidentified sources and of BSC giants as a function of M spectral type. The symbols "delta'' and "*'' refer to IRAS unidentified sources and BSC giants, respectively. The error bars shown on the data points correspond to the rms deviations of the colours from their mean values. The error bars on the mean colours of the BSC stars are shown slanted to avoid overlaps

  

Figure 5: [12]-[25] colour of IRAS unidentified sources which have best-quality flux density data vs. their M spectral type

  

Figure 6: Similar data as in Fig. 5 for BSC M giants

3.2 [12]-[25] colours of unidentified sources detected at both 12 and 25 ${\mu}$m

We have determined the [12]-[25] colours as a function of spectral type for all unidentified IRAS sources of quality-3 in this study, and these are presented in Fig. 5 for $\sim$1890 IRAS sources. One sees from the figure that there is an increase in [12]-[25] colour towards later spectral types and a large spread at each spectral type. We have used these data to obtain the mean [12]-[25] colours of these sources as a function of spectral type. To compare these data with those of M giants in the BSC, we have plotted in Fig. 6 the [12]-[25] colours of the BSC stars for which IRAS photometric data of quality-3 are available in these bands. In Table 3 we compare the mean [12]-[25] colours of BSC stars with those of unidentified IRAS sources. The entries in Table 3 are as those in Table 2 except that the colour referred to here is [12]-[25] instead of $B_{\rm j}$-[12]. The dependences of the mean [12]-[25] colours of these two types of sources as a function of spectral type are presented in Fig. 7. It is seen that the mean [12]-[25] colours of the IRAS sources are consistently higher than those of BSC stars as was the case in Fig. 4 although the colours (in the region of overlap) are within the statistical errors. K-S and Wilcoxon tests again indicate that the two samples are from different populations at the 95$\%$ confidence level. In Fig. 4, the difference was explained as due to greater interstellar extinction for the fainter, unidentified IRAS sources, whereas the difference between the two groups cannot be explained that way for the data in Fig. 7 since interstellar extinction is nearly negligible at both 12 and 25 ${\mu}$m. The difference may be a selection effect due to the difference in the volumes sampled around the faint, more distant IRAS sources with thick CSE compared to the local BSC M stars with almost dust-free photospheres.

  

Table 3: Mean [12]-[25] colours of M-type BSC stars and IRAS unidentified sources

An index of the variability of the source, listed as IVAR (expressed as a percentage) is available from IRAS PSC for 1890 unidentified IRAS sources which have quality-3 flux densities at both 12 ${\mu}$m and 25 ${\mu}$m and spectral type assignments. Of these 1334 have a variability index IVAR <25 and the remaining 556 have a variability index IVAR >25. The value of this index is based on correlated observations of IRAS sources simultaneously in the two spectral bands, 12 ${\mu}$m and at 25 ${\mu}$m. We present in Table 4 the distribution of these unidentified IRAS sources with [12]-[25] colours in two bins of IVAR. It will be noticed that about 25$\%$ of the sources with spectral types ranging from M3 to M7 have IVAR >25 but the percentage of sources with IVAR >25 increases sharply from spectral type M8 onwards to a value of about 55$\%$ at spectral type M10. This is however, not surprising as M stars of later spectral type are known to have a higher incidence of variability. We present in Fig. 8 the [12]-[25] colour vs. M spectral type for these sources whose IRAS variability index IVAR is <25 and, in Fig. 9, [12]-[25] colour for sources whose variability index IVAR is >25. The mean [12]-[25] colours of these sources as a function of their spectral type appear not to depend on the variability index (within the limits of errors on these mean colours). K-S and Wilcoxon two-sided tests indicate that there is about 90$\%$ chance that the points in Figs. 8 and 9 (are from the same population). We therfore conclude that variability has minimal effect on the distribution of points in Fig. 5.

  

Table 4: No. distribution of unidentified IRAS sources with [12]-[25] colours in two bins of IRAS variability index (IVAR)

  

Figure 7: Mean [12]-[25] colours of IRAS unidentified sources and BSC giants vs. M spectral type. The symbols "delta'' and "*'' have the same meanings as in Fig. 4. The error bars shown on the data points correspond to the rms deviations of the colours from their mean values. The error bars on the mean colours of the BSC stars are shown slanted to avoid overlaps

  

Figure 8: [12]-[25] colour vs. M spectral type of unidentified IRAS sources with quality-3 flux densities at both 12 ${\mu}$m and 25 ${\mu}$m and whose variability index IVAR is <25

  

Figure 9: [12]-[25] colour vs. M spectral type of unidentified IRAS sources with quality-3 flux densities at both 12 ${\mu}$m and 25 ${\mu}$m and whose variability index IVAR is >25

We point out that, the mean [12]-[25] colours of the northern IRAS sources classified by Stephenson (1986) for M6-M9, fall within 0.1 mag of ours which adds confidence to our conclusion.

3.3 [25]-[60] colours vs. spectral type of unidentified sources

The [25]-[60] colours of IRAS unidentified sources and BSC M giants as a function of spectral type are presented in Figs. 10 and 11, respectively. The mean [25]-[60] colours of the two groups are presented in Fig. 12, and the mean data are given in Table 5. The entries in Table 5 are as those in Table 2 except that the colour referred to here is [25]-[60] instead of $B_{\rm j}$-[12] as in the case of Table 2. The mean [25]-[60] colours of unidentified IRAS sources are seen to be always higher than those of BSC stars in the region of overlap of spectral types as in the case of $B_{\rm j}$-[12] and [12]-[25] colours. The K-S and Wilcoxon tests show that the IRAS sources are different from the BSC sources at the 95$\%$ confidence level.

  

Figure 10: [25]-[60] colour of IRAS unidentified sources which have best-quality flux density data in both these wavelength bands vs. their spectral type

  

Figure 11: Similar data as in Fig. 10 for BSC giants

  

Figure 12: Mean [25]-[60] colours of unidentified IRAS sources and BSC giants vs. M spectral type. The symbols "delta'' and "*'' have the same meanings as in Fig. 4. A large number of sources have [25]-[60] colours which deviate by a very large margin from their mean values (see Fig. 10). The mean [25]-[60] colours (shown in this figure) of the IRAS unidentified sources are therefore limited to only those sources whose [25]-[60] colour is $\leq$ 1.0. Mean colours of stars with spectral types numbering $\leq$ 5 are not shown in the figure as they are not statistically significant. The error bars shown on the data points correspond to the rms deviations of the colours from their mean values. The error bars on the mean colours of the BSC stars are shown slanted to avoid overlaps

  

Figure 13: Diagram showing the regions in the colour-colour plot of [25]-[60] vs. [12]-[25] that separate different types of stars with CSE (adopted from van der Veen & Habing 1988; VH). The colours [25]-[60] and [12]-[25] are as defined by VH. The evolutionary track is indicated by the curve "b'', $([25]{-}[60])=-2.15+0.35\ {\times}\ {\rm e}^{1.5([12]{-}[25])}$ which represents the observed colours very well for -1.1 < ([12]-[25]) <1.3. The line "a'' shows the loci of black bodies with temperatures ranging from 10000 K to 160 K with specific values shown at the vertical bars on the line. The circles refer to the positions of the IRAS unidentified sources of M spectral type

  

Table 5: Mean [25]-[60] colours of M-type BSC stars and IRAS unidentified sources