The Observability of Rotation Features

Thus far there are only a few examples of protostars known that exhibit features that could be caused by rotating disks. The main reason for this is that the disks tend to be quite small (100s-1000s of AUs) and hence are at the very limit of what present day telescopes can observe. However, there are several new telescope projects in progress at present that should deliver dramatically higher resolutions and sensitivities in the first decade of the 21$^{\rm st}$ century. It will therefore be instructive to simulate what features telescopes of differing resolutions can distinguish.

To do this consider a cloud that has the same envelope structure as that used for the testing of Buckley's model [7] in section 5.2.3. To recap, this was a 0.047 pc radius cloud with the inner 0.0175 pc collapsing primarily with a $r^{-\frac{1}{2}}$ velocity profile but with an additional $r^{0.15}$ profile adding to the velocity in the outer collapse regions. The molecular hydrogen distribution follows a $r^{-\frac{3}{2}}$ profile in the inner cloud, increasing to $r^{-2}$ in the outer cloud. Temperature follows $r^{-0.36}$ and the turbulent velocity and molecular abundance are constant throughout the cloud.

A disk is now placed in the centre of the cloud by making a disk shaped segment rotate with a Keplerian velocity distribution - ie. $c^{-\frac{1}{2}}$ where $c$ is the distance from the axis of symmetry in the cloud. Initially there will be no additional density enhancement in this disk (ie. the parameters in the disk are exactly those that the non-rotating cloud has except for the addition of rotation). The Keplerian velocity is defined by

$$v_t= \begin{cases} \frac{1}{10}c^{-\frac{1}{2}} \hspace{5mm}... ...vspace{3mm} \\ 0 & \text{$0.01<r<0.047$\space pc} \end{cases}$$

where $c$ is the distance perpendicular to the central axis of symmetry and $d$ is the distance from the central point of the cloud along the axis of symmetry. The other parameters, as before, are defined by:

$$v_r= \begin{cases} \left[ -0.49\left(\frac{r}{0.0175}\right)... ...pace{3mm} \\ 0 & \text{$0.0175<r<0.047$\space pc} \end{cases}$$

$$n_{\rm H_2}= \begin{cases} 2 \times 10^5\left[0.35\left(\fra... ...m} {\rm cm}^{-3} & \text{$0.0175<r<0.047$\space pc} \end{cases}$$

$$T_{\rm kin}=20\left(\frac{r}{0.0047}\right)^{-0.36} {\rm K}$$

$$V_{\rm turb}=0.705 \hspace{5mm} {\rm km s}^{-1}$$

$$\frac{n_{\rm CS}}{n_{\rm H_2}}=5 \times 10^{-9}$$

In addition the velocity is capped at a maximum of 8  ${\rm km \,s}^{-1}$. This may seem rather artificial but is necessary to stop the velocity climbing towards infinity at the centre of the cloud. It makes very little difference to the final result since only a tiny part of the central cloud has velocities higher than this.

  

Figure: Line profiles for HCO+ $4\rightarrow 3$ at offsets of 0 $^{\prime \prime}$, 0.1 $^{\prime \prime}$, 0.2 $^{\prime \prime}$, 0.5 $^{\prime \prime}$ & 1 $^{\prime \prime}$ for a cloud with Keplerian rotation capped at 8 km s$^{-1}$ (top) and Keplerian rotation peaking at the inner ring at 32 km s$^{-1}$ (bottom).

This can be seen in figure 6.5 where a pencil beam is used on two clouds, one with the rotation velocity capped at 8 km s$^{-1}$ and the other with the velocity increasing up to the innermost shell at $1\times10^{-5}$pc where the velocity peaks at 32 km s$^{-1}$. The line profiles are shown for offsets of 0 $^{\prime \prime}$, 0.2 $^{\prime \prime}$, 0.3 $^{\prime \prime}$, 0.5 $^{\prime \prime}$ & 1 $^{\prime \prime}$. Note that at a distance of 140pc $0.1^{\prime \prime}=7\times10^{-5}$pc$=15$AU and that the 8 km s$^{-1}$ mark is reached at approximately 0.00015 pc so it might be expected that only the two line profiles that are at 0.1 $^{\prime \prime}$ and 0.2 $^{\prime \prime}$ will show any significant differences and this is indeed the case. The lack of smoothness in these two line profiles is due to there being insufficient rings in the inner cloud leading to large differences in velocities from one ring to the next causing the step like effect. It can be seen that only line profiles very close to the centre of the cloud are affected by the cut off. In any realistically sized beam this problem is likely to be smeared out. It therefore seems valid to cap the velocity at 8 km s$^{-1}$. This rules out the possibility of unpredictable behaviour caused by using only 11 velocity steps to cover a velocity range of 30+ km s$^{-1}$ in the lambda iteration routine.

The general shape of the line profiles are as might be expected for the cloud with a significant line wing due to the rotation and a large self absorption dip at zero velocity caused by the cooler, stationary outer half of the cloud. The extra small dips and asymmetries, especially significant on the line profile that looked at the centre of the cloud, are due to the cloud being fairly optically thick which causes additional self absorption besides that caused by the outer regions of the cloud.

 

Line profiles for a 15 $^{\prime \prime}$ beam on the cloud centre with different numbers of gridding points. On the left is the entire line on the right is a blow up of the lower third of the line.

	nodd	no. of points
solid	201	40401
dashed	101	10201
dot-dashed	51	2601
dotted	21	441
dot-dot-dot-dash	11	121

This very steep rise in velocity at the centre of the cloud causes problems when trying to model the beam of the telescope as the very rapid changes in the cloud centre require large numbers of lines of sight to simulate the beam. Figure 6.6 shows the line profiles generated with varying numbers of gridding points (recall nodd is the number of gridding points on one side of the square within which the beam is sampled). It can be seen that the line wings are completely missed by the lower sampling rates and it is really necessary to use $nodd \gtrsim 100$ to get the correct line wings. This causes a significant reduction in speed (as might be expected) in the calculation of the line profiles. The line profiles in figure 6.6 were generated individually with the program run times shown below:  

nodd	run time (min)
11	0.15
21	0.37
51	1.9
101	7.4
201	29.2

 

Position velocity diagram for HCO+ 4 $\rightarrow 3$ using collision coefficients from S.Green [10] (top) and T.S.Monteiro [22] (bottom). The models are identical except for the collision coefficients.

Clearly it would be desirable to keep nodd to around 50 if possible and for all beams that do not cover the centre of the cloud around $nodd \approx 20$ would be adequate. A useful future modification to the program would be one that enables it to automatically select the best value of nodd depending on where in the cloud the beam is located. This would enable maps of the cloud to be produced much faster than at present. For now however, it is necessary to choose the value of nodd that is needed at the cloud centre and use it for all positions on the cloud. The values that will generally be used in the following models are either $nodd=51$ or $nodd=101$. A model producing line profiles at 50 positions across the cloud takes a little less than 1 hour for $nodd=51$ but around 3 hours for $nodd=101$. The time taken to produce a position velocity diagram depends not only on the number of grid points used to model the beam (and of course the number of rings in the cloud) but also the optical depth of the cloud. Calculating line profiles for an optically thick cloud can take much longer than for an optically thin cloud since each line of sight will then be split into many subsections when solving the radiative transfer equation (see page ).

A final problem that needs investigating are the collision coefficients. For HCO+ there are two sources of these collision rates. Either the rates of N$^2$H$^+$ as calculated by Sheldon Green [10] are used and multiplied by 2.5 as suggested by Monteiro [22] to simulate HCO$^+$ or the rates from T.S. Monteiro (private communication but based on work published in [22]) can be used. The former cover the correct temperature range for this kind of cloud, ie. 5-40 K but only cover the first 6 levels. The later cover the first 10 levels but unfortunately the temperature range is 20-250 K. The temperature range in this model cloud is 8.8-35 K so the outer parts of the cloud are well below the lower end of the range for which collision coefficients are given. Nonetheless when attempting to model high-J transitions there is no choice other than to use these coefficients. The problem is probably not too severe for two reasons. Firstly the high-J transitions will map mainly the hot parts of the cloud which in this case means only the central core and here the temperatures are close to or above 20 K. Also the collision coefficients do not change much with temperature. Therefore the interpolation routine which uses the collision coefficients at the lowest two temperatures to linearly extrapolate below the lowest available temperature will work fairly well if there are no dramatic changes between 20 K and 8 K (which there are not for most levels as can be seen from the S. Green rates). A check on this can be performed by comparing output from a model at J= $4\rightarrow 3$ using both sets of collision rates. This is shown in figure 6.7. It can clearly be seen that for all practical purposes these figures are the same and it is therefore possible to proceed with confidence to use only the collision coefficients that extend up to J$=10$.