Modelling L1544

The previous chapter has shown how it is possible to use the ASTRA program to predict the type of observations that might be possible for a given object. However, the main use for the program is more likely to be to help analyse data that has already been collected. There is only a fairly limited supply of objects that have been observed at a suitable resolution for this. The necessary resolution can be obtained with presently operating telescopes by considering objects that are as close to the Earth as possible. The closest star formation region of significant size is the Taurus molecular cloud. It contains several objects that have been studied in detail by various authors. The one that will be considered here is L1544. This has been studied over many years by P. Myers and collaborators (eg. [23]). They have concluded that it is a cold dense core that at present has no star in it but is "very advanced in its evolution toward the formation of one or more low-mass stars" (Tafalla et. al. [31]). They have observed it in a wide variety of molecules including CO, $^{13}$CO, C$^{18}$O, CS, C$^{34}$S, HCO$^+$, H$_2$CO, C$_3$H$_2$ and N$_2$H$^+$. The N$_2$H$^+$observations (see figure 7.2) were made with the Berkeley-Illinois-Maryland Array (BIMA) with a $14.8^{\prime \prime}\times 6.6^{\prime \prime}$ beam which at the distance of L1554 corresponds to 0.01$\times$0.0045pc (2000$\times$900 AU). This resolution is high enough to resolve a sufficient number of positions across the cloud that detailed modelling can be attempted. Model cloud used by Williams et al. [35]   [r] This has been done by Williams et. al. [35]. They performed radiative transfer modelling of the cloud and have been fairly successful in reproducing the observed line shapes. Their model consisted of two sheets of material collapsing onto one another as shown in figure 7.1. The front, low excitation layer falls towards the rear, high excitation layer producing the observed self absorbed line shapes. The two layers share the same thickness, kinetic temperature, turbulent velocity and density profiles but have different density peaks. The isodensity contours are taken to be elliptical in shape and there is a velocity gradient parallel to the major axis. They take the temperature to be constant at 12K across the entire cloud. The infall velocity is varied according to position to fit each spectrum individually. The turbulent velocity is taken to be 0.22 km s$^{-1}$, this too being held constant over the entire cloud. Across the plane of the cloud their hydrogen density is modelled by

$$n_{\rm H_2}=\frac{n_{\rm H_2,max}}{1+\left( \frac{x-x_0}{\Delta x} \right)^2+ \left( \frac{y-y_0}{\Delta y} \right)^2}$$

where $\Delta x$ and $\Delta y$ are the radii of the cloud in the $x$ and $y$ directions and $x_0$ and $y_0$are the co-ordinates of the emission peak. There is also a velocity gradient that runs parallel to the major axis in the cloud. This varies linearly at a rate of 3.8 km s$^{-1}$ pc$^{-1}$. The results of this model can be seen in figure 7.2. Clearly the model does provide quite good fits to most of the spectra although in several cases the size of the absorption dip is under estimated by the model. There is extremely good agreement for the peak intensity fitted on each spectra.

   

Figure: Spectra (histogram) and model fit (smooth line) from Williams et al. [35] for N$_2$H$^+$ emission in L1544.
Figure: BIMA map of the integrated intensity of N$_2$H$^+$ (101 $\rightarrow$012) emission in L1544 taken from Williams et al. [35].

Nonetheless, despite this reasonably good fit the model suffers from several deficiencies the most major of which is the lack of physical realism in the model. Clearly the cloud is unlikely to be made up out of two layers, it is much more likely to be a oblate or prolate spheroid. The model also makes no attempt to fit a global velocity structure to the cloud, instead it simply assigns a different infall velocity to each spectrum to make it fit the data and a velocity gradient that varies only with position along the major axis. Whilst both of these velocity structures provide a satisfactory fit to the data they are not based on any plausible physical model of the cloud.

Despite these criticisms the model successfully provides an approximate description of the conditions in this cloud. These can be used as a starting point for more sophisticated modelling using the ASTRA program. This is an ideal test candidate for the ASTRA program as it is clearly non-spherically symmetric and has proven difficult to model using traditional techniques. As a further complication the telescope beam and the cloud are not elongated in the same direction. The FITS data cube containing the data shown in figure 7.2 was kindly provided by Jonathan Williams and Philip Myers and will be modelled in detail in the next few sections.