## Statistics of the Velocity Field and Spatial Distribution of Hot Spots in Methanol Masers

### ABSTRACT

We present the results of the first study of statistical properties of velocity field and spatial distribution of the observed "hot spots" in methanol maser sources with available interferometric maps. Three Class I sources [DR21(OH), NGC6334IN, and L379] and one Class II source [W3(OH)] were investigated. In the majority of the sources, both the velocity difference between the pairs of spots and the average number density of the neighbors to a spot are adequately represented by a power law function of the spot separation. Both dependences can be interpreted as characteristic of a turbulent flow, although the total range of spot separations is not large enough to exclude other interpretations. The average fractal dimension of the observed spatial distribution of maser spots was found to be 1.02 ± 0.05 for the three Class I sources and 1.53 ± 0.29 for the Class II source. These low fractal dimensions may indicate that the underlying turbulence is strongly intermittent, but they may also result from unsaturated maser amplification in a homogeneous turbulent medium. Additional evidence is needed to distinguish between these two possibilities. The average slope of the velocity correlation function, measured over the whole range of spot separations, was found to be 0.65 ± 0.05 for the three Class I sources and 0.54 ± 0.05 for the Class II source. These values are significantly higher than the classical Kolmogorov value of 1/3 for the inertial sub-range of incompressible turbulence; this may indicate considerable dissipation of turbulent energy on all the scales probed by methanol masers.

### INTRODUCTION

Turbulence plays a fundamental role in astrophysics. It is an important feature of gas motion in many objects from the interplanetary gas in the Solar System through stellar atmospheres, through interstellar and intergalactic gas, to the pre-galactic gas of the early universe. One of the essential characteristics of typical astrophysical turbulence is its highly supersonic character. In contrast with Earth's atmospheric and oceanic turbulence, where the Mach numbers of turbulent pulsations are always much smaller than unity, the turbulence that arises in astrophysical objects generally has a Mach number in tens or hundreds. On Earth, we encounter supersonic turbulence of moderate Mach number in the case of supersonic flight, but the creation of highly supersonic turbulence in the laboratory for systematic study remains a very challenging task. Turbulent astrophysical objects may therefore present a valuable opportunity for studying this important phenomenon.

Astrophysical masers, and in particular, the masers observed in regions of active star formation, may cast light on the physics of supersonic turbulence. Typical maser sources appear as clusters of bright, compact "hot spots" with size roughly 10^{13} 10^{15} cm, spread over areas of characteristic scale of approximately 10^{16} 10^{17} cm. Walker (1984), Gwinn (1994) and Strelnitski et al. (2002) drew attention to the power-law dependence of the observed average velocity differences of 1.35 cm H_{2}O maser hot spots on the distance between the spots. A similar power-law dependence is known to characterize incompressible turbulence; this may indicate that H_{2}O masers in star forming regions are intimately connected with supersonic turbulence. Because of their brightness and compactness, H_{2}O masers may become a useful tool for studying the kinematic and geometrical properties of supersonic turbulence.

Methanol masers are also frequently observed in star forming regions. Methanol maser "hot spots" are about an order of magnitude larger than those of H_{2}O masers. While methanol masers demonstrate a narrower dispersion of velocities, the observed dispersions nonetheless indicate supersonic motions with Mach numbers between 4 and 8. If this velocity dispersion is due to supersonic turbulence, methanol masers may provide an additional opportunity for studying its properties. Unlike H_{2}O masers, for which the statistical properties of the spatial and velocity distributions have been investigated in prior studies (e.g. Strelnitski et al., 2002; Ripman, 2006), there has been no such investigations of methanol masers.

The goal of the present project was to cover this gap by statistically analyzing published interferometric maps of several methanol sources. Using the method of analysis described by Ripman (2006), we were able to demonstrate the feasibility of describing methanol maser statistics with power laws similar to those used in describing H_{2}O masers. We conclude that regardless of the larger dimensions and smaller velocity ranges of methanol maser spots, they may become a useful addition to water masers in studying supersonic turbulence.

### METHODS

We analyzed three Class I methanol maser sources in the 7_{0}-6_{1[/sub[A+ rotational line at 44 GHz: L379, DR21(OH) and NGC6334I(N). These three sources were chosen from the eight mapped with the VLA array by Kogan & Slysh (1998), because they had the largest number of measured maser spots and the largest spatial and velocity coverage. The spatial resolution of these observations was 0.2-0.5 arcseconds though most of the spots were unresolved with this resolution. The relative positions of the spots were measured with a higher accuracy of about 0.01 arcsec. The velocity resolution was 0.17 km/s. The spots within each of these sources typically occupy an area of several tens of arcseconds and cover a velocity range of 4-6 km/s.
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The ratio of the largest to the smallest measured spatial scales is typically 10 H^{2}-10^{3}, while the ratio of the largest to the smallest velocity increments is typically 10^{1}-10^{2}.

We also analyzed the prototypical class II source - W3(OH) - in the rotational line 2_{0}-3_{-1} E at 12 GHz. In this analysis, we used the VLBA data of Moscadelli et al. (2003) obtained with a high spectral (0.02 km/s) and spatial (~1 milliarcsec) resolution. These authors give relative coordinates and radial velocities for 43 maser spots, which are grouped into 3 clumps. In this paper we separately analyze the two clumps that contain the greatest number of maser spots namely, W3(OH)-1 and W3(OH)-2 containing 27 and 11 spots, respectively.

### THE GEOMETRY OF THE MASER SPOT DISTRIBUTION

We investigated the dimensionality of the spatial distribution of the maser spots in three Class I methanol sources using the density-radius method described in Strelnitski et al. (2002). Defining σ(r) as the average number density of maser spots at distance r from each maser in the region being studied:

A log-log graph of σ(r) versus r should be well-approximated by a straight line for objects of single-valued fractal dimension, with the slope of the line, d(log σ)/d(log r), giving the fractal dimension of the spot distribution according to Equation 2, where d_{0} is the dimension of the supporting space (d_{0} = 2 for a maser map).

The code used to calculate d according to Equations 1 and 2 was written by one of the authors (B.R.) and was successfully tested on a straight line an object of dimension 1. The code operates as follows: each maser spot is paired with every other maser spot and the distance r between these paired spots is calculated, producing a large number of data points. These data points are then binned according to their r-values, with the program keeping track of the number of data points falling within each bin. The program then divides the number of data points in each bin by the number of maser spots on the map and uses the results to plot σ(r) as a function of r. Finally, the program applies a best-fit line to the graph and calculates d in accordance with equation (2). When calculating d, we used bins of logarithmic width 0.25 in other words, the maximum r-value for data points placed within each bin was 10^{0.25} times larger than the maximum r-value for data points placed within the next-smallest bin.

An example of the graph of σ(r) vs. r is presented in Figure 1. The distribution of the points, covering about 1.5 orders of magnitude in r, is fairly well approximated by a straight line The RMS deviation of the points from the fitting line is about 10% of the interval of change of σ. For the determined slope, -0.84 ± 0.09 Equation 1 gives d = 1.15 ± 0.09. The values of d for other sources are presented in Table 1.

### STATISTICS OF THE VELOCITY FIELD

One of the fundamental properties of incompressible turbulence, predicted in a quantitative form by Kolmogorov (1941) and subsequently confirmed through experiment (Frisch, 1995), is the existence of the so called "inertial sub-range" of geometrical scales, η is greater than l is greater than L. Here, L is the outer scale at which turbulence is supplied with mechanical energy and η is the inner scale at which the mechanical energy of turbulence is dissipated through viscous effects. Within the inertial sub-range, energy cascades from larger to smaller scales due to the specific hydrodynamical instabilities and practically does not dissipate. Kolmogorov predicted that within the inertial sub-range the average absolute value of the velocity difference, Δv, between two points separated by a distance l is proportional to l^{1/3}. Experiments with incompressible turbulence have roughly confirmed this prediction.

It is believed that in the case of supersonic turbulence, shock waves produced at intermediate scales should dissipate considerable amounts of energy. It is easily shown that any loss of energy should steepen the slope of the velocity correlation function Δv(l). Soon after the creation of the first "classical" Kolmogorov theory, various investigators (including Kolmogorov himself) pointed out that intermittency of energy dissipation, which had been observed experimentally, should also steepen the velocity spectrum. Quantitatively, the predicted effect for incompressible turbulence is small and difficult to extract from the unavoidable experimental errors.

We analyzed the statistical properties of the 2D projection of the radial component of the velocity field in the same four methanol maser sources using the program "Structure Function Analyzer" (SFA; Ripman, 2006). SFA finds the slope of the log-log dependence of the average two-point velocity differences between maser spot pairs on the separation of the maser spots. The data points corresponding to individual spot pairs are placed into bins, averaged within each bin, and then plotted on a graph of log(Δv) log(l). In the case of a small number of mapped maser spots, the bin size Δl can be made very small, so that practically all the pairs of spots are treated individually. We used bins of size log(Δl) = 0.001. With the modest number of masers spots available, many bins of this small size turn out to be empty. The SFA ignores these empty bins when finding the least-squares best-fit line.

Once the preliminary best-fit line has been applied to the binned maser pairs, the next step is to fit the points on the log(Δv) log(l) graph with a second order polynomial, to see how much the observed log-log dependence deviates from linearity. If the deviation is not considerable, we limit ourselves with a single straight line fit for the whole range of spot separations. In the cases of strong deviations, three partly overlapping linear fits, for low, medium, and high separations are applied.

Examples of the determination of the spectral slope q for L379 and for the W3(OH)-1 cluster are presented in Figure 2. In the case of L379, the modest number of available maser spots and the form of the second order polynomial fit indicate that we should limit ourselves to a single straight line fit for the whole range of scales. In the case of W3(OH)-1, the number of points is sufficient to investigate the deviations of the log(Δv) log( l) dependence from a single-valued power law and the dependence of q on spatial scale. Examining the second-order polynomial fit and the three-range straight line fits, one can see that q decreases with increasing spatial scale, similar to what was found by Ripman (2006) in the case of water masers. The particular values of q are given in Table 1.

### DISCUSSION

This first attempt at studying the statistical properties of the velocity field probed by the methanol masers in regions of star formation and the dimensionality of their spatial distribution showed that their connection with turbulence cannot be excluded. In most cases, both the radial velocity difference between the pairs of spots and the average number density of the neighbors to a spot as functions of the spot separation can be approximated by power laws. The corresponding power indices are not dissimilar from those calculated in the case of water masers, (Ripman, 2006) and differ considerably from the classical values for incompressible turbulence.

In particular, much as in the case of water masers, the fractal dimension of the spatial distribution of maser spots (≈1) is much lower than that of intermittent incompressible turbulence (≈2.6). If methanol maser hot spots are physical condensations marking the sites of enhanced turbulent activity and/or dissipation, then the low fractal dimension simply indicates a high degree of intermittency. That is a low filling factor discribes the sites of high activity. However, numerical modeling of maser propagation in turbulent media (Holder et al., 2006, private communication) shows that unsaturated (exponential) amplification in a homogeneous, or non-intermittent, random velocity field can produce clustered distribution of hot spots on the output surface of the maser, and this can simulate an approximate power law on a log-log graph of σ(r) versus r. Distinguishing between these two explanations can be especially ambiguous when the covered range of scales is narrow. Additional confirmation will be needed to choose between these two possible interpretations.

The observed velocity spectrum of the maser spots more reliably indicates that they probe turbulent media. All of the investigated sources show a general increase of the velocity increments with increasing scale. At least for small and medium spatial scales, the linear approximation of the log(Δv) log( l) dependence is satisfactory, and its slope is sensible in the context of turbulence with considerable dissipation of energy at those scales. The large dispersion of the data points on the graphs used to determine the power law exponents (exemplified by Fig. 2) is an indirect confirmation that the turbulent velocity fields they characterize are intermittent - intermittency should give rise to large deviations of the distribution of the velocity increments at every given scale from Gaussian distribution. This is clearly seen in the case of water masers, (Strelnitski et al., 2002) and apparently, in the case of methanol masers as well. A more quantitative investigation of this effect in methanol masers is needed.

### CONCLUSIONS

1. In all four of the methanol maser sources studied in this project, the observed dependence of both the velocity difference between maser spot pairs and the average number density of the neighbors to a spot on the spot separation can be reasonably well-described by power laws, which may indicate that these methanol masers are connected with turbulent gas flows.

2. The low fractal dimensions (≈1) deduced from the power laws describing the spatial distribution of maser spots in these sources may indicate that the turbulence they probe is highly intermittent. However, this result can also be attributed to unsaturated maser amplification in a more or less homogeneous turbulent velocity field.

3. The slope of the velocity correlation function measured over the whole range of spot separations is ≈ 0.5-0.6 in most of the sources. This is higher than the classical Kolmogorov value of 1/3 for the inertial sub-range of incompressible turbulence and may indicate considerable dissipation of turbulent energy on all of the scales probed by methanol masers.

### ACKNOWLEDGEMENTS

The authors would like to thank Dr. V. Strelnitski for his mentorship and the staff of the Maria Mitchell Association for their assistance. This research was supported by the NSF/REU grant AST-0354056 and the Nantucket Maria Mitchell Association.

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