Observing Using the JCMT

 

$3 \rightarrow 2$ $4\rightarrow 3$

$9\rightarrow 8$ $4\rightarrow 3$

Position velocity diagram for HCO+ $3 \rightarrow 2$ (top left), $4\rightarrow 3$ (top right), $9\rightarrow 8$ (bottom left) for the JCMT and $4\rightarrow 3$ for a hypothetical giant telescope with an infinitesimal beam (bottom right). See text for details.

It is of interest to know how the line profiles of clouds similar to this one are affected by looking at different transitions with various different telescopes. The JCMT has receivers that can detect the J $=3\rightarrow 2$ and J $=4\rightarrow3$ transitions at resolutions of, respectively 18 $^{\prime \prime}$ and 14 $^{\prime \prime}$. There is also the potential for it to operate with a receiver that could detect the J $=9\rightarrow8$ transition at 802 GHz at which frequency it has a beamsize of 6 $^{\prime \prime}$. The position velocity diagrams for this model at these three transitions are shown in figures 6.8. Also shown for comparison in Figure 6.8 is the $4\rightarrow 3$ transition with a pencil beam. These diagrams show two problems associated with attempting to see such features with the present generation of telescopes. With a 15 metre dish such as that on the JCMT only at very high frequencies does the beam size become small enough to start revealing the signature of the disk rotation. However, at these high frequencies the emission from HCO+ is restricted to only a small area in the centre of the cloud which in turn restricts the amount of detail about the structure that can be seen simply because most of the disk has no emission. This problem is clearly visible in the $9\rightarrow 8$ diagram - note how weak the emission is here compared to that at the low frequency transitions. For HCO+ the problem is that there is a large gap between the $4\rightarrow 3$transition and the $9\rightarrow 8$ where all the HCO+ emission lines happen to fall in parts of the spectrum where the atmosphere is not transparent. It is thus not easily possible to find a compromise transition where the beam size is fairly small but most of the cloud is still excited. It is clear from the lower right diagram in figure 6.8 that the situation is much improved by using a telescope capable of high resolution at low frequencies. This is only really feasible with an array. Since most of the features of interest are fairly compact this problem is well suited to being studied with an interferometer especially since the problems of large structures being missed should not arise.

However, it would be useful to be able to make progress with smaller telescopes so it may be advantageous to consider using a different tracer molecule. By far the most commonly observed tracer molecule is Carbon Monoxide and its isotopes. Due to the numerous observable isotopes of differing abundances it is usually possible to choose an isotope that probes to the required depth in any given situation. The six most common isotopes are listed in table 6.1 together with their standard abundances. The standard abundances are derived assuming that $^{12}$C$^{16}$O has an abundance relative to H$_2$ of $1\times10^{-4}$ and taking the abundance ratios of the various isotopes from Gordy & Cook [9].

 

Table 6.1: The six most common CO isotopes and their standard abundances.

Isotope	Abundance
	relative to H$_2$
$^{12}$C$^{16}$O	$1.0\times10^{-4}$
$^{13}$C$^{16}$O	$1.1\times10^{-6}$
$^{12}$C$^{18}$O	$2.0\times10^{-7}$
$^{12}$C$^{17}$O	$3.7\times10^{-8}$
$^{13}$C$^{18}$O	$2.3\times10^{-9}$
$^{13}$C$^{17}$O	$4.2\times10^{-10}$

It will therefore be useful to run the model for a variety of different transitions for the isotopes of CO. The results from these runs are presented in figures 6.10 - 6.15. For each isotope the transitions that are potentially observable with the JCMT (given an appropriate receiver) are presented. Generally the $5 \rightarrow 4$ and $7 \rightarrow 6$ transitions fall in regions where the atmosphere is not transparent and are therefore not shown. The exception to this is the $7 \rightarrow 6$ transition of CO which is observable whereas the $8\rightarrow 7$ transition is not (it is the other way round for all other isotopes). The $1 \rightarrow 0$ transition is observable for all isotopes but for the JCMT has no advantage over the $2 \rightarrow 1$ transition due to a slightly worse atmospheric window and a significantly larger beam size. Also the C$^{17}$O $4\rightarrow 3$ transition is in an atmospheric absorption band and is therefore not observable.

 

Table 6.2: Beamwidths of JCMT for each transition of CO.

Transition	Beamwidth
$2 \rightarrow 1$	20 $^{\prime \prime}$
$3 \rightarrow 2$	14 $^{\prime \prime}$
$4\rightarrow 3$	11 $^{\prime \prime}$
$6\rightarrow 5$	7 $^{\prime \prime}$
$7 \rightarrow 6$	6 $^{\prime \prime}$

Position-velocity diagram for a non-rotating cloud in $^{13}$CO, $3 \rightarrow 2$.   [l] Figure 6.9 shows the position velocity diagram for a non-rotating cloud for comparison (all other parameters in the cloud are identical). The line profiles here show the standard double peaked signature of infall with the outer, cooler, cloud layers showing in absorption against the inner hotter regions. The symmetrical bulge in the position velocity diagram is also a signature of infall caused by the inner cloud having high velocity material causing the line wings.

For all the diagrams each line profile was calculated for 101 velocity points. The number of positions where the line profile was calculated varies according to the beam size. In all the diagrams a line profile was calculated every $\frac{1}{2}$beam-width. The beamwidths used for each transition are shown in table 6.2 (for all isotopes other than CO the $8\rightarrow 7$ transition was used rather than the $7 \rightarrow 6$ transition at 6 $^{\prime \prime}$). For each line profile calculated the number of gridding points used to simulate the beam was $51\times51=2601$. As can be seen from figure 6.6 and the associated discussion on page  this is not necessarily enough to correctly reproduce the extremities of the line wings. Nonetheless it is not practical to use a larger number as the run times then become unacceptably large. The result of this is that the lowest level contour in the diagrams is often not smooth - especially close to the centre of the cloud. This is due to the very centre of the cloud, where the rapid changes in velocity are taking place, being simulated differently depending on where the beam is centred. The general shape of the outermost contour is, however, assumed to be correct. This has been checked for a few diagrams by re-running them with $101\times101=10201$gridding points which virtually eliminates the problem.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$4\rightarrow 3$ $6\rightarrow 5$

$7 \rightarrow 6$

Position velocity diagrams for CO $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $4\rightarrow 3$ (middle left), $6\rightarrow 5$ (middle right) and $7 \rightarrow 6$ (bottom). See text for details.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$4\rightarrow 3$ $6\rightarrow 5$

$8\rightarrow 7$

Position velocity diagrams for $^{13}$CO $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $4\rightarrow 3$ (middle left), $6\rightarrow 5$ (middle right) and $8\rightarrow 7$ (bottom). See text for details.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$4\rightarrow 3$ $6\rightarrow 5$

$8\rightarrow 7$

Position velocity diagrams for C$^{18}$O $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $4\rightarrow 3$ (middle left), $6\rightarrow 5$ (middle right) and $8\rightarrow 7$ (bottom). See text for details.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$6\rightarrow 5$ $8\rightarrow 7$

Position velocity diagrams for C$^{17}$O $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $6\rightarrow 5$ (bottom left) and $8\rightarrow 7$ (bottom right). See text for details.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$4\rightarrow 3$ $6\rightarrow 5$

$8\rightarrow 7$

Position velocity diagrams for $^{13}$C$^{18}$O $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $4\rightarrow 3$ (middle left), $6\rightarrow 5$ (middle right) and $8\rightarrow 7$ (bottom). See text for details.

 

$2 \rightarrow 1$ $3 \rightarrow 2$

$4\rightarrow 3$ $6\rightarrow 5$

$8\rightarrow 7$

Position velocity diagrams for $^{13}$C$^{17}$O $2 \rightarrow 1$ (top left), $3 \rightarrow 2$ (top right), $4\rightarrow 3$ (middle left), $6\rightarrow 5$ (middle right) and $8\rightarrow 7$ (bottom). See text for details.

The position velocity diagrams in general show that the features that could be caused only by rotation are difficult to observe. The line wings in these diagrams are in fact fairly easy to observe, however they can also be caused by infall. It is asymmetric line profiles that point to rotation. Although the line wings here are significantly stronger than those shown in the non-rotating diagram in figure 6.9, increasing the infall velocity would also cause the wings to become larger. Thus unless some other means exists to determine the infall velocity the large line wings alone do not point to rotation. The asymmetry caused by rotation is unique and is caused by the opposite velocities being in the plane of the sky rather than along the line of sight as for infall (ie. one half of the plane of the sky has the cloud travelling towards the observer and the other half has the cloud travelling away from the observer - for infall it would be the same over the entire plane). Beyond detecting the asymmetry, being able to determine the rate at which the line wing decreases away from the centre of the cloud is necessary to be able to deduce the rotation curve which in turn yields the central mass of the object. This requires significantly better observations than just being able to detect asymmetry in the line shape.

Now considering each isotope in turn:

CO

The rectangular shape in the lower transition diagrams here is caused by the emission being in the optically thick LTE situation described in section 5.2.1. This causes flat topped line profiles across virtually all the cloud. The only exception is close to the centre where the high rotation velocities cause the emission to be spread out sufficiently so that it is no longer optically thick at the higher velocities, leading to the line wings. This only occurs for those beams that cover the central few arcseconds of the cloud. This effect reduces for the higher transitions and by the $6\rightarrow 5$ transition has disappeared alltogether. These higher J transitions are promising candidates for observing as they are producing bright lines that would be observable with the inevitably high noise receivers at these frequencies and yet are not so optically thick that all the information from the central regions of the cloud is lost. Additionally of course the higher frequencies mean a much higher resolution. CO is also unique in being the only isotope to have the $7 \rightarrow 6$ transition observable in the 800-900 GHz atmospheric window. All the other isotopes have their $8\rightarrow 7$ transition in this window which produces lines that when combined with the reduced abundances are more than an order of magnitude weaker. This makes this transition unique in that it is the only transition at for which the line wings would be easily detectable at the highest resolution at which the JCMT can work. If line profiles could be obtained with noise levels of only a few $\frac{1}{10}$ of Kelvin then the velocity profiles in the cloud should be determinable from this diagram.

$^{13}$CO

The $2 \rightarrow 1$ transition is the only transition for this isotope that suffers from the excessive optical depth problem that many of the CO transitions have. The self absorption visible in nearly all the CO transitions is only significant in the $2 \rightarrow 1$ and $3 \rightarrow 2$ transitions here. The $6\rightarrow 5$ transition especially shows a significantly different line shape near the cloud centre for this isotope. The $8\rightarrow 7$ transition would be very difficult to detect as it peaks at only 0.4 K and in any case contains little information as it is only present in the very centre of the cloud where the highest densities and temperatures exist. The $3 \rightarrow 2$ transition here can be compared directly with that shown in figure 6.9 which is the same cloud with no rotation. The main differences are a slightly weaker maximum and more pronounced asymmetric line wings. The difference in the wings is only detectable at fairly weak levels though (consider the two 0.5 K contours) demonstrating that detecting rotation features requires high quality data. Only CO and $^{13}$CO have sufficient line strength at the $4\rightarrow 3$ and $6\rightarrow 5$transitions to be able to measure the rotation curve of the cloud. The higher resolutions at these transitions are necessary to be able to measure the drop off in the line wings but in the less abundant isotopes these transitions are then too weak.

C$^{18}$O

Large changes in the lineshapes are apparent here when compared to the previous two higher abundance isotopes. There is no sign of self absorption in any of the transitions which would make this a very useful isotope for obtaining line profiles at the centre that are relatively easy to interpret without having to consider optical depth effects. In other words this is the first transition that is probing all the way to the centre of the cloud. The abundance is, however, now low enough that the $8\rightarrow 7$transition so weak that it is no longer detectable. Even the $6\rightarrow 5$ transition, whilst easily detectable, now contains little information off the cloud core.

C$^{17}$O

Only 4 transitions for this isotope are likely to be detected from the ground as the $4\rightarrow 3$ transition falls in an atmospheric absorption band and the $8\rightarrow 7$ transition is too weak to be detected leaving only the $1 \rightarrow 0$, $2 \rightarrow 1$, $3 \rightarrow 2$ and $6\rightarrow 5$. Only long integrations on the $3 \rightarrow 2$ transition would start to show up the asymmetric line wings signifying rotation at the 0.1 K level. In the $2 \rightarrow 1$ transition they are probably not detectable.

$^{13}$C$^{18}$O

The effects of the low abundance are obvious here in that there are now insufficient molecules to produce significant strength lines at any transition. The $2 \rightarrow 1$ and $3 \rightarrow 2$transitions would certainly be detectable but the line wings signifying rotation would not be detectable.

$^{13}$C$^{17}$O

With very long integrations this may just be detectable at the central position in the $2 \rightarrow 1$ and $3 \rightarrow 2$ transitions, however the emission is too weak to be able to gain any information from the line shape. The higher transitions are presently undetectable, although at a very low level they retain the same shapes as for the equivalent $^{13}$C$^{18}$O transitions.

 

CO $1 \rightarrow 0$ $^{13}$CO $1 \rightarrow 0$

CO $2 \rightarrow 1$ $^{13}$CO $2 \rightarrow 1$

CO $3 \rightarrow 2$ $^{13}$CO $3 \rightarrow 2$

Position velocity diagrams for CO (left column) with transitions for $1 \rightarrow 0$ (top left), $2 \rightarrow 1$ (middle left) and $3 \rightarrow 2$ (bottom left) and $^{13}$CO (right column) with transitions for $1 \rightarrow 0$ (top right), $2 \rightarrow 1$ (middle right) and $3 \rightarrow 2$ (bottom right) for the LMT 50 metre telescope. See text for details.

 

C$^{18}$O $1 \rightarrow 0$ C$^{17}$O $1 \rightarrow 0$

C$^{18}$O $2 \rightarrow 1$ C$^{17}$O $2 \rightarrow 1$

C$^{18}$O $3 \rightarrow 2$ C$^{17}$O $3 \rightarrow 2$

Position velocity diagrams for C$^{18}$O (left column) with transitions for $1 \rightarrow 0$ (top left), $2 \rightarrow 1$ (middle left) and $3 \rightarrow 2$ (bottom left) and C$^{17}$O (right column) with transitions for $1 \rightarrow 0$ (top right), $2 \rightarrow 1$ (middle right) and $3 \rightarrow 2$ (bottom right) for the LMT 50 metre telescope. See text for details.