Tsallis Distribution and ISM Turbulence: A New Way of Constraining Mach Numbers

Benjamin M. Tofflemire1, Blakesley Burkhart2, Alex Lazarian2

1University of Washington, 2University of Wisconsin

Simulation Data SetWe generate a database of 14 three dimensional numerical simulations (5123

resolution) of isothermal compressible MHD turbulence by using the Cho & Lazarian (2002) code and varying the input values for the sonic and Alfvénic Mach number.The sonic Mach number is defined as,

Ms ≡ <|v|/C

s>

Where v is the local velocity and Cs is

the sound speed. Averaging is done over the whole simulation. Similarly, the Alfvénic Mach number is,

MA ≡ <|v|/v

A>

where vA=|B|/ρ½ is the Alfvénic velocity, B

is magnetic field, and ρ is density.

Analysis of 2D Column DensityWhile the analysis of our 3D density simulations using Tsallis statistics is valuable in its own right as a tool to study MHD turbulence, analogous observational data is unobtainable. Accordingly, we create synthetic 2D column density maps from our simulations to test Tsallis on observable quantities. Figure 1 display a rendering of a highly turbulent and highly magnetized simulation.

This same exercise was carried out for smoothed, noise added, and cloud bounded synthetic column densities as well as PPV cubes in an effort to recreate observable data. The Tsallis statistic was able to describe these PDFs well with consistent sensitivities M

s and M

A. This makes Tsallis a potentially valuable tools for observational

analysis in synthesis with other statistical tools.SEE OUR PAPER COMING SOON, Tofflemire et al. 2011, ''Interstellar Sonic and Alfvénic Mach Numbers and the Tsallis Distribution,'' ApJ,

Submitted!!!

ReferencesCho, J., & Lazarian, A., 2002, Physical Review Letters, 88, 245001Esquivel, A., & Lazarian, A., 2010, ApJ, 710, 125Shivamoggi, K. B., 1995, Annals of Physics, 243, 177Tsallis, C., 1998, Journal of Statistical Physics, 52, 479Tofflemire B. M., Burkhart, B., Lazarian, A., 2011, ApJ, Submitted

AcknowledgementsAuthors wish to thank A. Esquivel. B.T. is supported by the NSF funded Research Experience for Undergraduates (REU) program through NSF award AST-1004881. B.B. acknowledges the NSF Graduate Research Fellowship and the NASA Wisconsin Space Grant Institution. We also thank the financial support of the Center for Magnetic Self-Organization in Astrophysical and Laboratory Plasmas.

PDFs show three trends. PDFs become more Gaussian with decreased M

s,

Gaussianity increases with lag for supersonic cases, and Gaussianity increases with M

A.

In the search for sensitivities to Ms and

MA, we plot fit parameters a, q, and w

(presented as w--2) in Figure 3 vs spatial

lag for all 14 simulations. For a and q, solid lines plot average values for supersonic, transonic, and subsonic. For

q, only supersonic and subsonic average lines are plotted.

Parameters a and w--2 show strong

sensitivities to Ms. w--2 display the

strongest magnetic sensitivity as shown

in Figure 4. Here w—2 is plotted for the high (red diamond) and low (blue triangle) magnetizations of 2 supersonic (top), 1 transonic (bottom left), and 1 subsonic (bottom right) simulation. In each case the higher B produced a higher value. Figure 4

Figure 3

MotivationUnderstanding the dynamics and evolution of the interstellar medium (ISM) is critical to advancing our knowledge of many astrophysical phenomena such as star formation, cosmic ray physics, magnetic reconnection, galaxy evolution, and magnetic dynamo theory. Astrophysical MHD turbulence plays a major role in each of these phenomena and occurs on many energy injection scales ranging from kpc down to sub-AU.

Subsonic

Supersonic

Tsallis StatisticsThe Tsallis distribution (Tsallis 1988) was originally derived to describe fractal and multifractal systems. These systems apply to many natural environments such as ISM turbulence (Shivamoggi 1995). Esquivel & Lazarian (2010) used Tsallis statistics to describe the spatial variation in probability distribution functions (PDFs) of density, velocity, and magnetization of MHD simulations similar to those used here but of lower resolution.The Tsallis function of an arbitrary incremental PDF Δf has the form:

Where Δf ->

The fit is described by the three dependent parameters a, q, and w. The a parameter

describes the amplitude while w is related to the width or dispersion of the distribution.

Parameter q, referred to as the ''non-extensivity parameter'' or ''entropic index'', describes the sharpness and tail size of the distribution.

Figure 1

Figure 2

Unfortunately, direct observations of ISM turbulence are extremely difficult. They most often come from statistical analysis of 2D column density maps where the physical characteristics are largely unknown. MHD simulations provide an alternative avenue for research in the quest for tools to describe turulence. Here we test a new tool, Tsallis statistics, in an effort to characterize the sonic and Alfvén Mach numbers of ISM turbulence.

We investigate Tsallis fits of 2D column density maps by creating incremental PDFs for varying spatial separations. Fits and PDFs are shown in Figure 2 (red symbols are the data from the simulations, and the blue lines represent the Tsallis fit).