The Topology of Neutral Hydrogen in the Small Magellanic Cloud

Jeremy Gordon
University of Georgia
skeletor AT uga DOT edu

2007 University of Wisconsin REU Program
Astronomy Department

This summer, I worked with Dr. Chepurnov and Dr. Lazarian, analyzing an HI column density map of the Small Magellanic Cloud (SMC). Using a technique called genus statistics, we were able to determine the general topology of the SMC at different scales.

The Small Magellanic Cloud

The Small Magellanic Cloud is an irregular dwarf galaxy that is in orbit around the Milky Way galaxy. It's coordinates are RA 00^h52^m, Dec 72^o51", making it visible only from the southern hemisphere. It is roughly 60kpc distant, making it an excellent source in which to study the ISM and star formation in nearby galaxies. The image on the left is a 100 micron image, taken by IRAS. The image on the right was taken in the optical. Although they look quite different, there are similarities: the bright region white and blue regions in the IR are matched in the visual by the bright emission. It turns out that this emission is caused by young stars ionizing the hydrogen gas, as well as heating up the surrounding dust grains.[1]

Genus Statistics

The technique known as Genus Statistics is relatively new in the astronomical field. It was originally developed in the mid 1980s as a way to study the 3D topology of galaxies from the CfA survey. Later work led to the 2D genus, which is what we used in this project. The two-dimensional genus is defined below:

G = (isolated high-density regions) - (isolated low-density regions)

Take a C.D. for example. Projecting it onto a 2D grid (or simply looking down on it), we see a actual disc with a hole in the middle. In this case, the genus for the C.D. would be zero (one contiguous high-density region - one contiguous low-density region). Likewise, when looking down upon a frisbee, the genus would be +1 (one high-density region - one low-density region). This is the gist of how the genus statistic works.

The genus curve for a random Gaussian 2D distribution is known. This is important, as any deviation from this curve can tell us topological information for the selected region. The feature we focused on was the genus shift which can be roughly defined as the value at which the genus curve crosses the x-axis. It turns out that a genus curve that crosses the x-axis at the origin has a Gaussian distribution, or there are equal numbers of holes and clumps. A genus curve that crosses the x-axis to the left of the origin has a negative genus shift. This implies that there are more isolated clumps than holes, resulting in a negative genus shift.

Above is a toy image created in a 2007 study by Kim & Park. The solid line is the genus curve for a random Gaussian dsitribution of matter with a smoothing scale of 2'. The dotted line with closed circles represents the genus curve for a random distribution of matter with 857 extra clumps, while the dotted line with open circles represents the genus curve for a distribution of matter with 857 extra holes. This graph succintly explains the rationale behind our study. The dashed line with closed circles crosses the x-axis slightly to the left, implying a negative shift and a clump topology. Similarly, the dashed line with open circles crosses the x-axis slightly to the right, implying a positive shift and a hole (or swiss-cheese) topology.

This is the rough basis of our research. We analyzed various regions within the SMC (seen and described below) and plotted the genus shift vs. smoothing scale. This can tell us how the topology changes at different physical scales.

HI Column Density Map

The HI column density map used in this project is a composite, created with data from the Parkes Telescope as well as ATCA, a radio interferometer. The two data sets were merged to create the HI column density map seen above. The angular resolution of the combined beam is ~90", but to make a cleaner image the data was placed three pixels across. While each individual pixel has an angular resolution of ~30”, the effective angular resolution of the combined column density map is 88.56”, implying a spatial resolution of 25.76pc at a distance of 60kpc.

We cropped the SMC into nine overlapping 150x150 pixel regions in an attempt to minimize the inclusion of regions with zero column density. The genus statistic was run with a smoothing scale < 10% of the image length. For instance, the smoothing scale of the 150x150 region increased in increments of one pixel, up to a smoothing scale of 15 pixels.

We also cropped out a 400x300 pixel region of the SMC. This was the largest region of the SMC we felt we could safely crop out, minimizing the regions of zero column density surrounding the SMC. The results are shown below.

Results

The results from the 'entire' SMC are shown above. We are looking at a plot of Genus Shift vs. Smoothing Scale (in pc) for the selected 400x300 region. Immediately, a few interesting features appear. At small smoothing scales (26-75 pc), we see a decidedly negative genus shift. This indicates that the clumps dominate, resulting in the 'meatball' topology seen. As the smoothing scale increases (75-125 pc), the genus shift trends upwards, indicating that the clumps are coalescing and the shells are coming into focus. At these medium scales, a neutral/slightly positive genus shift is seen, indicating a weak hole topology. At large scales (>125 pc), the genus shift returns to a slightly negative value. It appears that at the largest scales explored, the clumps slightly outnumbers the holes, resulting in the slightly negative genus shift seen.

The results from the nine 150x150 regions are similar, yet different (these nine graphs can be seen in the attached paper below). At small scales (0-25 pc), each region exhibits a significant negative genus shift, indicating a clump topology at these scales. Similarly, as the scale increases (25-75pc), the genus shift steadily climbs to a neutral/slight positive genus shift, depending on the region in which you look. The real difference comes when looking at scales > 75pc. While the genus shift for the entire SMC dropped, these regions are slightly different. Some continue to rise, some stay steady, while others dip back down into negative values. This result shows that the SMC is inhomogeneous from region to region, and although the entire SMC shows a clump topology at large scales, the individual regions may not.

It is important to note that the 150x150 regions were probed to a maximum smoothing scale of 128.85 pc (15 pixels), while the 400x300 region was probed to a maximum smoothing scale of 257.7 pc (30 pixels). Therefore, the small regions show a slightly different genus shift curve, which is what we would expect.

Summary

We analyzed the HI column density of the SMC in an attempt to elucidate its topological features and concluded that the topology of the HI does indeed change at different scales. The summary of results are as follows:

At small scales (26-75 pc) the SMC exhibits a significant negative shift, indicating a clump or "meatball" topology. We conclude that this is due to the numerous clumps of gas and dust embedded within the SMC. At these small scales, the larger shells and SGSs do not contribute much to the genus as they are larger than the smoothing scales used.
As the scale increases (75-125 pc), the shift of the genus curve becomes less negative, trending towards a neutral/slight positive shift. This can be attributed to the larger shell and SGS structures throughout the SMC. At these medium scales, the smaller gas clumps coalesce together while the shells come into focus. These shells are most likely a result of stellar winds and SNe from OB associations.
For larger regions with scales > 125pc the genus curve becomes noisier; however, the same general trend can be seen. The genus shift levels out at a slightly negative value, indicating the presence of more isolated clumps as compared to shells. This result shows that despite the fact that shells are scattered throughout the SMC, clumps still dominate, even at the largest of scales.
The nine 150x150 pixel regions of the SMC exhibit slightly different trends. Although they all possess a clump topology at small scales, the curves at larger scales are vastly different. Some trend towards a hole topology at large l while others exhibit no positive genus shift. We can conclude from these individual regions that the SMC is quite heterogeneous from region to region. While the entire SMC may exhibit a clump topology at small scales and a hole topology at medium scales, individual regions will vary from one to another.
Genus analysis is an effective complementary tool in the study of turbulence. The power spectrum contains information on velocity fluctuations but does not possess topological information. Combined with genus statistics, both the velocity fluctuations and topological information can be obtained for a selected object.

The paper I worked on this summer can be found here. With any luck, it will be published within the next year or so. A more in-depth analysis of genus statistics and our methodology/results can be found here.

My presentation for the summer can be found here. It provides a similar overview that this website contains.

References

Gott, J. R., III, Dickinson, M., & Melott, A. L. 1986, ApJ, 306, 341

Kim, S., & Park, C. 2007, ApJ, 663, 244

Melott, A. L., Cohen, A. P., Hamilton, A. J. S., Gott, J. R. I., & Weinberg, D. H. 1989, ApJ, 345, 618

Multiwavelength Small Magellanic Cloud - Irregular Galaxy

Stanimirovic, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J., & Snowden, S. L. 1999, MNRAS, 302, 417