FEATURE ARTICLE
The Formation of Star Clusters
Clouds in the summer sky provide clues about the organization of star populations
Bruce Elmegreen, Yuri Efremov
Hierarchical Interstellar Gas
Over the past decade it has become apparent that much of the gas between stars is itself arranged in a hierarchical structure. This structure is evident across a wide range of gas-cloud sizes, from the smallest clumps that can be resolved with a telescope to giant clouds in the Galaxy's spiral arms. The large clumps of gas contain smaller clumps, most of these contain even smaller clumps, and it seems likely that this trend continues below the present limit of telescope resolution. This hierarchical structure is even present in clouds that are not forming stars, so it has little to do with the star- formation process itself. A clue to the cause of this structure comes from a closer examination of its properties.
As it happens, the hierarchical structure of interstellar gas clouds is approximately self-similar for a wide range of scales: The largest clouds are divided into smaller clumps in the same way as the smaller clumps are divided into even smaller clumps. In this respect, interstellar gas clouds are fractal, with a well-defined fractal dimension. The fractal dimension of a hierarchical structure is equal to the logarithm of the average number of subclumps inside each clump, divided by the logarithm of the size ratio of the large clump to the average small clump.
Consider a solid object such as a cube. It is three-dimensional, and its fractal dimension is also three. We can see this by dividing a cube into eight subcubes, with the side of each subcube half the length of the original cube. So there are two subcubes along each width, two along each height, and two along each depth, for 2 x 2 x 2 = 8 total subcubes. The logarithm of eight divided by the logarithm of two equals the fractal dimension of three. Or we could divide it into 3 x 3 x 3 subcubes (similar to a Rubik's cube), giving log 27/log 3 = 3, for the fractal dimension again.
If the partitioned cube is not completely filled, but only five of the eight subcubes are filled and the rest empty, then the fractal dimension is log 5/log 3, which is about 2.3. Now we can make a bigger hierarchy with this fractal dimension by subdividing each filled subcube into eight more and choosing another five of these for further subdivision, and so on down to arbitrarily small scales. The fractal dimension of this whole structure would still be 2.3.
Many attempts have been made to assess the fractal dimension of interstellar gas clouds. A recent study by one of us (Elmegreen), with Edith Falgarone of Paris, used the relative number of clouds and clumps of various sizes to derive a dimension of 2.3. (The example above was not chosen arbitrarily!) This result might not be otherwise interesting to astronomers, except that it is remarkably similar to the fractal dimension of structures seen in the earth's atmosphere, such as the wispy and swirling plumes in jet trails and atmospheric clouds. Indeed, billowy white clouds in the summer sky look a lot like star-forming clouds.
On earth, the self-similar structures in smoke plumes and atmospheric clouds are the result of turbulence. Turbulent motions produce hierarchical structures because the large-scale parts of the motion produce the large-scale concentrations, and the small-scale parts of the motion produce the small-scale concentrations.
Many people think of turbulence as the random jittery motions that toss airplanes around in an unpredictable fashion. But there is also a regular and systematic element to turbulence: Nearby regions tend to move at about the same speed, whereas distant regions move more independently. Very simply, the greater the separation between two fluid elements, the greater the difference in their relative velocities.
Most earth-bound turbulence has a pattern of velocities that increases as the cube root of the separation. This pattern appears to be generally true for incompressible fluids, and for disturbances in air that are at such low pressures that the density remains fairly constant. The relation is a little steeper for astronomical turbulence, with velocity scaling approximately as the square root of the separation. The difference may be the result of larger pressure fluctuations for astronomical turbulence. These fluctuations tend to be so large in space that they actually compress the gas, forming density enhancements. These are presumably the same density enhancements that make newborn star clusters.
The important point is that the velocity structure in turbulent regions of space is scale-free: The same physical processes and the same velocity-separation relations occur over a wide range of absolute scales, from hundredths of a light-year to thousands of light-years. The large velocities on large scales compress the gas into large clumps, and the small velocities on small scales compress the gas into small clumps. If the velocity structure in space is scale-free, then the density structure that results from these motions should be scale-free as well.
We believe this is the reason for the hierarchical structure of gas clouds in space, and for the analogous structure of the birth places of stars. Star clusters of all sizes are formed by the conversion of gas into stars, following the compression of gas into hierarchically distributed clumps in a turbulent interstellar medium.
The link between the hierarchical structure of gas clouds and open clusters is evident in the distribution of their masses. Hierarchical gas clouds have an intrinsic mass distribution in which the number of clouds with masses in a fixed interval M is proportional to 1/M2. So there are 100 times as many clouds with masses between 9 and 10 solar masses as there are clouds with masses between 99 and 100 solar masses, which is a factor of 10 larger. This mass distribution turns out to be the same as the distribution of masses for open clusters, a value we were able to establish in a recent study of large and small clusters in the Large Magellanic Cloud.
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