In my spare time I enjoy playing with fractals, those beautiful objects from theoretical mathematics. There are two kinds of fractals, deterministic and random fractals. The Mandelbrot set is the most commonly seen deterministic fractal. I focus much of my time working with a type of random fractal called fractional Brownian motion because it seems to describe the geometry of interstellar clouds quite well.
The Mandelbrot Set |
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For an excellent description of the Mandelbrot set, click here
To download my IDL code that you can use to make images of the Mandelbrot set, Click here. |
Fractional Brownian Motion
Zuzana Gedeon, of the Montana State University Department of Computer Science, described fractional Brownian motion succinctly.
Mandelbrot's fractal geometry has found a wide variety of applications.
The statistical self-similarity of fractal shapes is inherent in the natural
world and may be exploited to simulate natural processes. In particular,
random fractional Brownian motion (fBm) corresponds to fractal landscapes
and occurs widely in nature, accurately describing such phenomena as pitch
variation in music, flicker noise in solid-state devices, and 2-D mountain
landscapes.
FBm is simply a sum of randomly phase-shifted sine waves, the amplitude
of which varies with frequency as 1/fß for 1<=ß<=3.
FBm has a jagged trace which resembles the skyline of a mountain range.
Mandelbrot observed this and reasoned that extending the function to two
(-point-something, if you insist) dimensions would result in a surface
resembling mountainous terrain. He did so and presto! fractal mountains
were born.
Click here to download my IDL code that you can use to make images like the ones below.
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Mountains
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Clouds
Instead of a surface plot, plotting the pixel values as an image gives structure mimicking clouds. Of course, you have to chose the correct power law index ß. (A sky blue to white color table helps.)
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Interstellar Clouds/Starforming Regions
A different value for ß gives an image that looks like a typical starforming region. The technique can be
expanded to three spatial dimensions, plus a separate cube for radial velocity to simulate real spectral lines. This method is described in the Miville-Deschênes et al. reference (below).
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References
Qian, H. et al, "On two-dimensional fractional Brownian motion and fractional Brownian random field" 1998 J. Phys. A: Math. Gen 31(28) L527. Discusses the mathematics of 2D fBm.
Stutzki, J. et al. "On the fractal structure of molecular clouds" 1998, Astron. & Astrophys., 336, 697. Excellent discussion of fBm structures in the interstellar medium. Explains the relationship between the various fractal and other power law indices used to describe interstellar structure. Lots of detailed image-processing theory, with practical tips on how to simulate fBm images. Very useful.
Miville-Deschênes, M.A., Levrier, F., & Falgarone, E., "On the use of fractional Brownian motion to determine the 3D
statistical properties of interstellar gas" 2003, ApJ, 593, 831. This paper picks up where Stutzki left off, constructing cloud density and velocity fields using 3D fBm models, and "measuring" the statistics an astronomer would observe in spectral line data cubes.
