We have calculated the line ratios between the observed HC3N, HCN, and HNC lines, and the used method will be demonstrated in this appendix. The method accounts for beam size/source size effects, using the following formula for a line ratio between the spectral lines A and B: (A.1)where θmb is the main beam half-power beam width (or main beam size) of the telescope, ηmb is the main beam efficiency of the telescope, θs is the source size of the observed object, and is the integrated intensity of the signal. The temperature can also be given in Tmb scale – in this case, the main beam efficiency corresponding to that observation should be omitted since .
Some of the galaxies have very large angular distributions. The measured intensities might in these few cases not represent a global value for molecular gas in the galaxy, but rather a value for a certain (central) region of the galaxy. This will be discussed in Sect. A.1. If the different observations for the same galaxy are made in different parts of the galaxies the line ratios might be misleading. The exact positions of observation are given in most of the references, and these have been compared. The largest difference in position between two observations in the same galaxy is for IC 342, which has a 3′′ difference between the HCN observation in Sorai et al. (2002) and the HNC and HC3N observations in Henkel et al. (1988). Compared to the sizes of HNC and HC3N distribution (Meier & Turner 2005) as well as the used beam sizes (19′′ and 25′′), this offset is although rather small. For all other galaxies, the position difference is at most 1′′.
Another issue is that two molecules compared in a line ratio might not have the same spatial distributions, and too narrow a beam might exclude more flux from one molecule than from the other. The problem gets even greater if the two molecular lines are observed with two different beam sizes. The problem mainly affect the most nearby galaxies in the sample, whose source sizes are comparable with the used beam sizes.
The size of the dense molecular region is also important when converting from measured temperature to brightness temperature, and as a consequence of this also when computing the line ratios between two transitions in a galaxy when different telescopes have been used. When available, the source size has been estimated from tables or maps showing the HCN source size, thus assuming that the HCN source size is similar to the source sizes of the other molecules used in the line ratio calculations (HNC and HC3N) as all these lines are expected to be present only in the dense molecular regions (Meier & Turner 2005). When an HCN source size has not been found, the source size of the CO emission has been used instead. Preference has then been given to the higher CO transitions, as these are better tracers of dense gas than CO 1-0, which traces much thinner molecular gas, giving too high a value on the source size. The source sizes used in this work are shown in Table 3, and unless no transition is stated, HCN 1-0 should be assumed.
For some of the galaxies no reliable value on the source size (neither HCN nor CO) has been found. This might still not be a problem when calculating the line ratios. In Sect. A.2 it will be shown that the error when assuming a point source is less than 5% for objects more distant than 45 Mpc, and less than 10% for objects more distant than 30 Mpc. These errors are calculated for a line ratio made with the most different beam sizes used in this work. When the beam sizes of the telescopes used for the observations are (almost) the same, the source sizes will just cancel. In any case, this error should be smaller than errors introduced by the use of so many different telescopes and instruments.
As mentioned above, some of the most nearby galaxies have source sizes larger than the beam size of the telescope used for the observation. When calculating molecular line ratios, this might pose a problem. Only if we expect the distributions of the two molecules to have the same shape and size, and the observations are made with the same beam sizes, we will achieve the same ratio as for a global measurement on the galaxy. In particular, we expect the HC3N to be concentrated in a smaller region of the galaxy than HCN and HNC (Meier & Turner 2005), which makes the calculations of these ratios depend on the beam size to be large enough to cover the whole dense molecular region of the galaxy (e.g. the whole HCN region).
If the beam size is smaller than the HCN (or HNC) emitting region, but larger than the HC3N region, the HC3N/HCN (or HC3N/HNC) ratio will be overestimated, as all HC3N will be seen, but not all HCN (or HNC). However, if the opposite would be true, the HC3N/HCN and HC3N/HNC instead would be underestimated. But since HCN and HNC are more abundant than HC3N in all studied sources, this seems very unlikely.
Not knowing the HC3N source size will also affect the line ratios from the more distant galaxies to some extent, since the source size used for the line ratio calculations is an HCN source size (in a few cases even a CO source size) also for the HC3N intensities. However, assuming the proportion between the HCN and HC3N source sizes to be similar for all galaxies, this will affect all line ratios in the same way, thus making all line ratios a little bit too high.
Another problem for the line ratios of the nearby galaxies is that the two different observations used to calculate a line ratio sometimes are made with different beam sizes. When comparing HCN 1-0, HNC 1-0, and HC3N 10-9 observations made with the same telescopes the difference in source size is negligible, but if a line ratio is calculated from observations from two different telescopes the two molecular intensities are observed towards regions with different sizes. For distant galaxies this is not a problem, since the whole molecular region of the galaxy is unresolved in any beam. For the more nearby galaxies, one of the molecules might be observed more or less globally in the galaxy, while the other is observed very locally in the galactic centre, giving an erroneous line ratio. The galaxies affected by this issue should be the same as those affected by the previously mentioned beam size issue.
The source sizes used in this study are found in Table 3. However, for some sources no reliable value for the source size has been found. We will here discuss why this is not always a problem, and estimate sizes of the errors inflicted from not knowing the source size.
As can be seen above, the line ratios depend on the source and beam sizes with the factor . The source size θs can be ignored if both beam sizes θmbA,θmbB ≫ θs. If the beam sizes θmbA ≈ θmbB, the source size will also cancel. However, if θs is comparable to the beam sizes, and θmbA and θmbB are non-similar, the source size becomes an important factor.
The error when assuming the source to be point-like, e.g. setting θs = 0, will be (A.2)For the sources where an HCN source size has been used, the corresponding source diameter has never exceeded 1.6 kpc (NGC 2146). For the CO source sizes, the largest is found in NGC 5135, with a source diameter of 4.1 kpc in CO 1-0, but the HCN source sizes should be smaller than this.
Assuming no larger HCN source diameter than 1.6 kpc, and the largest and smallest beam sizes at 90 GHz (SEST with 57′′ and IRAM with 28′′), the error will be less than 5 % for distances greater than 45 Mpc, and less than 10 % for distances greater than 30 Mpc. Thus, even for objects closer than 30 Mpc, the error caused by the point-like approximation (θs = 0) is notable only if the θmb of the two observations used to calculate the ratio have notably different sizes (at least a 30% difference in beam sizes is needed to produce an error of 10 %).
© ESO, 2011