Keplerian Rotation

John Richer and Rachael Padman [27] have attempted to model optically thin disks that are in Keplerian rotation. That is, they have assigned a velocity field that depends only on radius and is given by

$$v_{\rm kep}(r)=\sqrt{\frac{GM^{\rm eff}_r}{r}}$$ (5.7)

where $M^{\rm eff}_r$ is the effective mass that when distributed spherically would cause the velocity $v_r$. Of course the mass is not distributed spherically but is concentrated at the star in the centre and the rest is distributed in the disk. However, for a infinitely thin disk this is the same as the true $M_r$ so as long as the height of the disk is not too great this discrepancy can be ignored. They then proceed to show that the line emission from such a disk can be given by

$$I_\nu(x,v)=2\int^\infty_{r=\vert x\vert}\varepsilon(r)\phi_\n... ...v-\frac{x}{r}v_{\rm kep}(r)\right]\frac{r}{\sqrt(r^2-x^2)}\,dr$$

where as in previous sections $\phi_\nu$ is the normalised Gaussian line shape, $v_{\rm kep}(r)$ is as given in equation 5.7, $\varepsilon(r)$ is the emissivity function, $r$ is the distance from the rotation axis of the cloud and $x$ is the impact parameter.

 

Optically thin disk rotation (taken from [27])

Using this equation they model the $^{13}$CO emission from a disk that is believed to be similar to the disk around HL-Tau. Figure 5.32 is the figure they present in their paper. The graph at the top left of the figure shows the distribution of emissivity that they used, ie. maximum emission at the very centre of the disk, dropping to near zero at the outer edge. The modelled cloud line profiles are presented in the form of a position-velocity diagram in the lower two diagrams. In a position-velocity diagram the line profiles are shown as contours. To find the line profile at a particular position, $x_1$, in a cloud find that position on the x-axis and then the line profile is made up from the values of the contours along the line $x=x_1$. This is an extremely compact way of showing large amounts of data. In figure  5.32 the lower left diagram is the output for a point sized beam and the lower right diagram simulates what would be seen by a beam of the size shown in the lower right corner.

These diagrams can be reproduced by the ASTRA program by having it produce output at many positions in the cloud. These can then be fed into the posvel program which produces a position-velocity diagram. In order to reproduce this model the following parameters were used. The parameters vary only with radius in the disk, there is no variation in the direction perpendicular to the disk. The height of the disk is not important but it has been chosen here to be large enough so that the beam is always filled (if this were not the case the only effect would be to lower the intensity of the lines). The cloud was assumed to be at a distance of 140 pc.    

$$v_r=0 \hspace{2mm},\hspace{2mm} v_z=0 \hspace{1cm} 0<r<0.02 pc$$

$$n_{\rm H_2}=1 \times 10^6 \hspace{5mm} {\rm cm}^{-3} \hspace{5mm} 0 < r < 0.02 pc$$

$$T_{\rm kin}=20 \hspace{5mm} K$$

$$V_{\rm turb}=0.25 \hspace{5mm} {\rm km \, s}^{-1}$$

$$\frac{N_{^{13}CO}}{N_{\rm H_2}}=2 \times 10^{-8} e^{-7r^2}$$

As Richer & Padman do not precisely describe the model they were using it is not possible to reproduce it in all its details. For example the precise form of the Gaussian used for the emissivity is not described, hence the rather arbitrary looking factor of 7 that appears in the $\frac{N_{^{13}CO}}{N_{\rm H_2}}$ equation above. However the general shape of the output should be reproducible with these parameters and indeed it is as can be seen in figure 5.33. This figure assumes a pencil-like beam (ie. a beam of infinitely narrow resolution) and was produced by sampling at $\frac{1}{2}^{\prime \prime}$ intervals. Each line profile was calculated at 151 velocity steps - this large number makes it easier to get a smooth looking position velocity diagram and does not significantly slow down the output (once the lambda iteration has been solved it only takes a few seconds to produce an output since there is no beam to be modelled. Note that the thick line represents the velocity profile in the cloud and is terminated at the edge of the cloud. The velocity is capped at 8 km s$^{-1}$ to avoid excessive velocity differentials in the centre of the cloud. From the innermost cylinder it declines linearly to zero at the centre (hence the slightly diagonal vertical line in the centre of the figure).
 

ASTRA reproduction of an optically thin rotating disk (18 cylinders)

Ring miss problem   [l] Producing these outputs with pencil beams (ie. only one line of sight used for each beam) revealed a problem with the program in the outer regions of the cloud that should be noted. The problem is obviously visible in figure 5.33 where the contours have a rather jagged shape. The cause of this can be examined in a slightly more exteme case shown in figure 5.35. This figure shows output at a series of points in a cloud. The first 3 and the last 3 lines are a series 1 arcsecond apart on the cloud (at 23 $^{\prime \prime}$, 22 $^{\prime \prime}$, 21 $^{\prime \prime}$,20 $^{\prime \prime}$, 19 $^{\prime \prime}$, 18 $^{\prime \prime}$), the middle 4 lines are 0.2 $^{\prime \prime}$ apart at 20.2 $^{\prime \prime}$, 20.4 $^{\prime \prime}$, 20.6 $^{\prime \prime}$ & 20.8 $^{\prime \prime}$. It can clearly be seen that the rise in intensity of the lines is not linear with position on the cloud. Consideration of figure 5.34 shows the reason for this: it turns out to be a slightly different manifestation of the problem that was described on page . The problem arises when a line of sight passes close to a cylinder boundary (in this type of model the parameters are invariant with vertical height so the disks play no role). If it is a line similar to line 'a' then all of its emissivity will be caused by the emissivity of cylinder 1. A line slightly further in or out than line 'a' will have its emissivity changed only very slightly due to the change in line length in cylinder 1. This is what is occuring for the first three lines in figure 5.35 where only a very slight upward trend is observed for the lines closer to the centre of the cloud. The right most three lines in figure 5.35 can also be seen to have a slow upward trend for line closer to the centre of the cloud but at a considerably higher intensity than the first three lines. This is due to ring 2 have a higher emissivity function than ring 1. These are therefore lines similar to line 'c' in figure 5.34. It now becomes clear why the middle 4 lines display such a rapid increase in intensity. They are equivalent to line 'b' and are very close to the edge of cylinder 2. However, when close to the edge of the ring small movements towards the centre of the cloud cause large increases in the length of the line of sight in cylinder 2 (and corresponding decrease in the length of the line in cylinder 1) and since cylinder 2 has a higher emissivity than cylinder 1 the line strength increases rapidly. In other words this problem is being caused by the function for emissivity (ie. the density and abundance distributions) not being smooth enough. This was exactly the same problem as with the velocity steps as described on page . Unfortunately the solution used there is not possible here. This is because previously the velocity information was stored as the geometries of the lines of sight were calculated. It was therefore possible to add another segment to the line of sight with the exact velocity at the closest point to the centre of the cloud stored. This is not the case for the density and abundance information and by the time this information is needed the absolute position of the line of sight relative to the centre of the cloud is no longer known. There is therefore no easy solution to this problem without re-writing significant sized sections of the program. However, it is possible to virtually remove the problem by increasing the number of cylinders used in the model. In this way the density and abundance distributions are smoothed out. As might be expected this is easier for some models than for others. Figure 5.33 was produced using a model with 18 cylinders. Increasing this to 48 yields figure 5.36 which removes virtually all signs of the irregularities. However, Richer & Padman also show a model for a cloud with emission coming from a Gaussian ring (see upper left panel in figure 5.37) where many more cylinders are necessary to produce a smooth output.
 

Example output at 23 $^{\prime \prime}$, 22 $^{\prime \prime}$, 21 $^{\prime \prime}$, 20.8 $^{\prime \prime}$, 20.6 $^{\prime \prime}$, 20.4 $^{\prime \prime}$, 20.2 $^{\prime \prime}$, 20 $^{\prime \prime}$, 19 $^{\prime \prime}$, 18 $^{\prime \prime}$ offset from the cloud centre.

 

ASTRA reproduction of an optically thin rotating disk (48 cylinders)

Modelling emission as a Gaussian ring is intended to reproduce either clouds which exhibit depletion of the trace molecules in their centres or which have an actual cavity in their centres. Figure 5.36 shows the modelled position velocity diagram for this cloud as published by Richer & Padman. Although they use this distribution as an example for modelling a much more massive cloud (30 M$_\odot$) the model reproduced here will have exactly the same parameters as the previous model in order to enable comparison with the simpler disk model described above. The position velocity diagram should however remain very similar. The ASTRA version of this model can be seen in figure 5.38. Due to the emissivity changing much more rapidly within the cloud many more cylinders are needed to produce smooth looking models. For example 180 cylinders were used to produce figure 5.38. It is also worth noting here that although it might be thought that 180 cylinders would dramatically slow the program down this can be compensated for by reducing the number of disks to a minimum - in this case just 4 were used. This is possible because of the invariance of the parameters with disk height. As a final test figures 5.39 & 5.40 are produced using exactly the same parameters except this time a 15 $^{\prime \prime}$ beam was used. These two figures are intended to approximate the two diagrams presented by Richer & Padman showing the line profiles convolved with a beam.

  

Optically thin disk rotation (taken from [27]) with emission in a Gaussian ring (see text for details). ASTRA reproduction of an optically thin rotating disk (180 cylinders).

Despite the minor differences in the models used, the general reproduction of the models is very close to the originals in figures 5.32 & 5.37. The small differences present are probably mainly due to different values being used for the contour lines. The contour lines given by Richer & Padman are arbitrary but linearly spaced and thus it is difficult to exactly reproduce them. The contour lines in the ASTRA models are also linearly spaced but it is unknown what the linear spacing should be, nor where the first contour should lie. It should also be noted that the model reproductions with a beam are not strictly the same as those presented by Richer & Padman as there is no velocity convolution. The beam was modelled as a JCMT type beam (ie. 15 $^{\prime \prime}$) but the velocity steps were kept the same, this may explain the larger discrepancies for the beam models.

  

ASTRA reproduction of an optically thin rotating disk with a 15 $^{\prime \prime}$ beam. ASTRA reproduction of an optically thin disk rotation with emission in a Gaussian ring with a 15 $^{\prime \prime}$beam.