D RAFT VERSION F EBRUARY 2, 2018 Preprint typeset using LATEX style AASTeX6 v. 1.0
DIAGNOSTICS OF TURBULENCE PROPERTIES THROUGH VELOCITY GRADIENT AMPLITUDES: THEORY AND OBSERVATIONS K A H O Y UEN 1 ,V ICTOR L AZARIAN 1 , A. L AZARIAN 1
arXiv:1802.00024v1 [astro-ph.GA] 31 Jan 2018
1
Department of Astronomy, University of Wisconsin-Madison
ABSTRACT Based on the properties of magneto-hydrodynamic (MHD) turbulence, the recent development of the Velocity Gradient Technique (VGT) allows observers to trace magnetic field directions using spectroscopic information. In this paper, we explore how the amplitudes of gradients from different observables are correlated to the properties of turbulence in terms of statistical moments. We find there is a very good power-law relationship between the dispersion of gradient amplitudes and sonic Mach Number Ms , and the increase of alignments of the gradient amplitudes filaments to the decrease of Alfvenic Mach number MA . The visual and quantitative relation between the shapes of gradient amplitudes to Mach numbers allows us to easily evaluate physical conditions. We illustrate our method in GALFA-HI data and find that the high latitude regions have higher Ms but lower MA , which requires theoretical explanations. Keywords: ISM: general — ISM: structure — ISM: magnetic field — magneto-hydrodynamics (MHD) — radio continuum 1. INTRODUCTION
Turbulence is ubiquitous in the universe. Together with the magnetic field, magneto-hydrodynamic (MHD) turbulence plays a very important role in various astrophysical phenomena (see Armstrong et al. 1995; Chepurnov & Lazarian 2010), including star formation (see McKee & Ostriker 2007; Mac Low & Klessen 2004), propagation and acceleration of cosmic rays (see Jokipii 1966; Yan & Lazarian 2008), as well as regulating heat and mass transportation between different ISM phases (see Draine 2009 for the list of the phases). The importance of MHD turbulence was not fully revealed until the recent theoretical breakthrough in the understanding of MHD turbulence by Goldreich & Sridhar (1995) and later theoretical and numerical works (Lazarian & Vishniac 1999; Cho & Lazarian 2002, 2003; Kowal et al. 2009). Supported by the advancement of MHD turbulence theory, the idea of using spectroscopic velocity gradients to trace magnetic field, namely the Velocity Gradient Technique (VGT, Gonz´alez-Casanova & Lazarian 2017; Yuen & Lazarian 2017a,b; Lazarian & Yuen 2018), was recently proposed and widely applied to observation data from HI to selfgravitating molecular clouds. The orientations of velocity gradients are taken into account for tracing both the direction of magnetic field in diffuse (Yuen & Lazarian 2017a) and self-gravitating media (Yuen & Lazarian 2017b; Lazarian & Yuen 2018) along with field strength (Yuen et al in prep) based on turbulence scaling with the concept of blockaveraging (Yuen & Lazarian 2017a). The vast application of VGT illustrates its capability to trace magnetic fields in different regimes.
[email protected]
In a separate development, the notion of gradients (both orientations and amplitudes) are widely used in sparse and theoretically loose frameworks. On one hand, studies involving intensity gradient orientations suggested that density structures are aligned perpendicularly to the magnetic field based on magnetization (Soler et al. 2013) or local flow (Soler & Hennebelle 2017). A similar analysis of filamentary structures in velocity channel intensity (Clark et al. 2015) revealed there is an empirical relation between intensity structures within channel maps and magnetic fields. In fact, earlier theoretical studies already predicted that thin channel maps (and their thick variant, intensity maps) contain information about turbulent eddies (Lazarian & Pogosyan 2000, henceforth LP00, Lazarian & Pogosyan 2004, 2006). On the other hand, the amplitudes of polarization gradients are suggested to be correlated to the nature of turbulence, especially the sonic Mach number (Gaensler et al. 2011; Burkhart et al. 2012). However, the statistics of gradient amplitudes in the framework of VGT are only addressed in YL17b when considering the regime of shocks. As previously mentioned, observing the orientation itself can already provide both field orientations and strength, i.e. MA . Amplitudes can reflect the compressional nature of turbulence. Compared to method of density PDF (which suggests σ 2 ∝ log(1 + A2 Ms2 ), see Federrath et al. 2010; Burkhart & Lazarian 2012), using statistics of velocity gradient amplitudes with the concept of block averaging allows one to estimate the local sonic Mach number. To outline this paper, we first describe our numerical setup in §2. We analyze the gradient amplitudes in 3 and illustrate our method in observation maps in §4. We discuss some potential issues and prospects in §5 and summarize our work in §6.
2 2. TESTS WITH SYNTHETIC OBSERVATIONS
The numerical data is the same as what we used in YL17b and LY17. For details of how we set up the simulation using the 3D MHD Eulerian code ZEUS-MP/3D, please refer to the two papers above. Our simulations employ various Alfvenic Mach numbers MA = VL /VA and sonic Mach numbers Ms = VL /Vs , where VL is the injection velocity, VA is the Alfven velocity and Vs is the sonic velocity, which are listed in Table 1. The regime MA < Ms corresponds to the case when the plasma magnetic pressure is larger than the thermal pressure, i.e. plasma with low β/2 = Vs2 /VA2 < 1, while the regime MA > Ms corresponds to the pressure dominated case with β/2 > 1. Set
Model
Ms
MA
Resolution
B
b11 b12 b13 b14 b15 b21 b22 b23 b31 b32 b41 b42 b51 b52
0.4 0.8 1.6 3.2 6.4 0.4 0.8 1.6 0.4 0.8 0.132 0.264 0.04 0.08
0.04 0.08 0.16 0.32 0.64 0.132 0.264 0.528 0.4 0.8 0.4 0.8 0.4 0.8
4803 4803 4803 4803 4803 4803 4803 4803 4803 4803 4803 4803 4803 4803
Table 1. Simulations used in this work, the Ms and MA are measured at driving stage.
Figure 1. Visual illustration on how Ms changes |∇I|, |∇C| and
∇P .
The raw data from simulation cubes are converted to synthetic maps for our gradient studies. Assuming the line-ofsight direction is the x-axis, the intensity I(r) and normalized velocity centroid C(r) are defined as: Z I(r) = ρ(r, x)dx Z (1) C(r) = I −1 ρ(r, x)vx (r, x)dx Gradients are computed following the procedure proposed in Yuen & Lazarian (2017a). Both intensity gradient amplitudes (|∇I|) and velocity centroid gradient amplitudes (|∇C|) are calculated by taking the modulus. We note that our approach has similarities to the one employing polarization gradients (Gaensler et al. (2011), Burkhart et al. (2012)) on inferometric data, which is based on the sum of gradient amplitude of Stokes parameters Q and U: 1 ∇P = (∇Q2 + ∇U 2 ) 2 (2) Correlation was found between polarization gradients and fluid compressibility (see Burkhart et al. 2012). 3. ANALYZING GRADIENT AMPLITUDES
3.1. Visual comparisons of gradient amplitudes to Ms and MA We will first illustrate visually how the structures of gradient amplitudes are correlated with the dimensionless physical parameters. Fig 1 shows how |∇I|, |∇C| behave when Ms is changing by keeping MA approximately constant. First of all one can easily observe that the width of filamentary structures is inversely correlated with Ms for all three cases. This suggests a very simple way of estimating local Ms by just visual comparison between the regions. Second, the widths of |∇C| are generally smaller than that of |∇I|. The most likely reason is that the amplitudes of intensities are contributed to by both turbulent motions and shocks, while the amplitudes of velocities and magnetic field strengths are mostly contributed by turbulence motions only. Third, as MA decreases, the filaments straighten out. This can be explained by the velocity centroid structures being mostly parallel to the magnetic field in case of small MA , but beginning to deviate in the case of large MA (See Gonz´alez-Casanova & Lazarian 2017; Lazarian & Yuen 2018). However, the straightening of intensity structures are less apparent as that of |∇C|, which is expected in earlier literature (Esquivel & Lazarian 2005).
3
Figure 2. The linear structures of |∇C| and |∇I| by varying Ms (Left panel) and MA (Right panel). The color shows the enhancements of linear structures in the |∇C| space. The width and orientations of these filaments with respect to a number of cubes with different Ms and MA show a clear relationship between the structures we visually see in Fig 1 and the quantitative information, filament width and en masse alignment.
3.2. Extracting filamentary structures from gradient amplitude maps As there are obvious filamentary structures laying inside these windows, we use the Rolling Hough Transform (RHT, Clark et al. 2014), which is a linear structure detection algorithm specially designed for extracting filamentary structures1 . We then use the window length of 65 pixels, mask level of 0.7, and smoothing radius of 15 pixels (See Clark et al. 2014 for the meaning of the parameters) for our case of filtering. Figure 2 shows 6 snapshots of RHT-processed |∇C| and |∇I| with varying Ms and MA . One can, again, easily see
1 Readers have to be careful that, the original RHT implementation is for analyzing the linear structures in Velocity Channel Maps (See Clark et al. 2015). We borrowed the idea of filament extraction but not necessarily agree that filaments in channel maps represent any actual density structures. On the contrary, the theory describing Position-Position-Velocity (PPV) statistics in LP00 suggest that velocity crowding effects produce filaments in thin channel maps even the underlying density is set to be constant (e.g. (Yuen & Lazarian 2017b; Lazarian & Yuen 2018) for a detailed discussion).
that the visual thickness and orientation of filaments are correlated to both Ms and MA . On one hand, when we decrease Ms while keeping MA constant, the thickness of gradient amplitudes increases, especially for |∇I| compared to |∇C|. On the other hand, if we decrease MA , the linear structures tend to be straighter and oriented in parallel to the magnetic field (which all three cases in Fig 2 the mean field is pointing right). We can easily quantify some of the features in the filament maps using statistical tools. The simplest way of quantifying the thickness of the filaments in the amplitude space would be using normalized statistical dispersion σ/µ because, if the structure of the map is in general thick, most of the regions will be zero. We use the dispersion of filament intensities σ divided by the mean value of intensity µ as well as the angle θ between the filament and the horizontal axis. The upper row of panels of Figure 3 shows the dependences of σ/µ and low row of panels illustrates the dependences of the mean angle θ. The expression σ/µ shows the most clear empirical dependence on Ms . For intensity filaments the dependence
4
Figure 3. Comparisons between the normalized standard deviation σ/µ and mean alignment of filaments of both |∇I| (red) and |∇C| (blue)
against Ms (Left) and MA (Middle) and β ∝ (MA /Ms )2 (Right). For the cases with obvious linear relation, we provide the fit with the fitting parameters drawn respectively.
is
σ ≈ 1.5 + 0.2Ms µ
(3)
and for the velocity centroids filaments the dependence is σ ≈ 1.5 + 0.3Ms (4) µ which is expected both visually from Fig 2 and from previous experiences with the method based on the variance of densities (Burkhart & Lazarian 2012). While the normalized dispersion of |∇I| is in general larger than that of |∇C|, they both exhibit a linear relation to the sonic Mach number Ms . However, we do not observe a linear relation between σ/µ and MA , while the dependence on the plasma magnetization β shows the decrease of the ratio as beta increases. Interestingly enough, for the mean angle θ the dependence on the sonic Mach number Ms is not well defined for the entire range of Ms studied, while hθi increases as β increases. At the same time the dependence on the Alfvenic Mach number for centroids and obeys an empirical relation shown in the middle lower panel. The plot of |∇I| is significantly higher than that of |∇C| even though the slopes of their fitting lines against MA are approximately the same. Thus hθi and σ/µ are complementary measures that can be used to find Ms and MA . The two linear relations (σ/µ − Ms , hθi − MA ) show that the filamentary structure of gradient amplitude can provide both information of Ms and MA in a direct and simple way, which is very advantageous for observers. We also notice that |∇I|, in general, has larger dispersion and deviations in all six plots shown in Fig 3. This is expected as density information is inherited from velocities and is influenced by processes in which GS95 scaling is not ap-
plicable, for example the occurrence of shocks in high Ms environments, which tend to increase the dispersion of densities and also its gradients. 4. APPLICATIONS TO NEUTRAL HYDROGEN DATA
To demonstrate our method (§3), we use the GALFAHI (Galactic Arecibo L-band Feed Array HI) survey of the Galaxy in the 21cm neutral hydrogen line obtained with the Arecibo Observatory 305 meter telescope, whose large aperture has an angular resolution ∼ 40 . We illustrate our technique using the same region that is used in Clark et al. (2015) for the studies of filaments. The corresponding information from the data release can be found in Peek et al. (2018). The region covers the sky at right ascension (R.A.) across 263.5◦ −196.6◦ , and declination (Dec.) across 22.5◦ −35.3◦ . For reference, this piece of data covers a wide range of galactic latitude from middle to high, which is 26.08◦ − 83.71◦ , where the lower left region indicates low galactic latitude, and upper right region represents high latitude. We illustrate our technique by (1) Compute |∇I| after obtaining the intensity map. (2) Perform RHT filament extractions on |∇I| (3) Compute the statistics in the amplitudefilament space. On top of Fig 4 shows our result of amplitude filaments and the predicted sonic Mach number. We then compare the dimensionless normalized standard dispersion σ/µ to obtain an estimation of Ms from Fig 3, and obtain an estimate of Ms in the bottom of Fig 4. We also use the method of column density variance (Burkhart & Lazarian 2012) to obtain the relative scale of Ms within their framework. For both methods, we use the block sampling size of 1002 pixels for local comparison. We do see a number of features unique to this region based
5
Figure 4. (Top) The method of column density variance (Burkhart & Lazarian 2012) for this piece of data in relative scale. (Middle) The
amplitude filament map of |∇I| from a piece of GALFA-HI region in Clark et al. (2015). (Bottom) The predicted sonic Mach number based on the filaments’ normalized standard deviation.
on our visual inspection of numerical cubes. On the right hand side of the region, the structures of intensity gradient amplitudes are linear and thin, which indicates the high latitude part of the sky might have high sonic Mach number but low Alfvenic Mach number compared to the lower latitude region. We compare our method to the method using variation of logarithm density (Federrath et al. 2010; Burkhart & Lazarian 2012) and find similar results in the high latitude region. Moving from high latitude to lower latitude, the filamentary structure becomes thicker and more entangled, which indicates a decrease of sonic Mach number and an increase of Alfvenic Mach number. As on the left hand side of the region, the filaments of intensity gradient amplitudes are completely misaligned, close to what we have in Fig 2 for MA ∼ 1. The thickness of filaments in low-latitude regions are also the greatest. We find that the higher latitude clouds might have higher Ms and lower MA . Our result on Ms differs from that obtained using the variations of column density in Burkhart & Lazarian (2012). We suspect that for the latter case, noise contribution may be more important compared to the technique that we employ here. The reason behind is because the noises cannot be smoothed in the density PDFs. As a result, the number of points required by the density PDF technique has to be much larger than that for the gradient technique,
which prohibits observers to perform studies on local Ms . 5. DISCUSSION
5.1. Importance to the Velocity Gradient Technique The Velocity Gradient Technique (Gonz´alez-Casanova & Lazarian 2017; Yuen & Lazarian 2017a,b; Lazarian & Yuen 2018) in its current form provides a reliable way of tracing magnetic field directions based on the modern understanding of MHD turbulence. Our work here is the first to connect the gradient amplitudes from VGT to both Mach numbers and provide a first numerical relation between gradient amplitude properties to them. The visual effect from gradient amplitude filaments is also very handy for observers to approximate the physical conditions. The prospect of using gradients to probe physical conditions significantly increases the value of the Velocity Gradient Technique when tracing magnetic field directions, which also helps observers to further constrain the physical conditions in different regimes and might help to rule out some of the competing models of star formation (See Crutcher et al. 2010; Lazarian et al. 2012). 5.2. Complementary nature to other methods studying Ms and MA The complementary nature of using the gradient amplitude method to both variance of logarithm of column density
6 (Burkhart & Lazarian 2012) and the Chandersakkar-Fermi method (Chandrasekhar & Fermi 1953; Falceta-Gonc¸alves et al. 2008) for polarization angle are also advantageous to the observer. While logarithm of column density and column density gradient amplitudes are closely connected to each other, the unique ability for using gradient amplitudes (both column density and velocity centroid) to predict MA is not provided in the variance method in Burkhart & Lazarian 2012. However, we also show the centroid provides a more closely bounded and relatively less disperse parameter which is advantageous for application to observational data with significant noises since the gradient operator tends to amplify noises. This work, which is a variant of the Velocity Gradient Technique, provides the first estimate of Sonic and Alfvenic Mach number (and also compressibility of turbulence) in space independently from polarimetry in a cost-effective way2 . Synergistic use of these methods provides a way to cross-check the measurements. Moreover, in case when polarimetry data is unavailable, the gradient technique provides an alternative way to study the magnetization of interstellar media. 5.3. A higher resolution study through the gradient amplitude method Inferometry is not limited to polarimetry measurements. It is possible to measure inferometric information with a much higher resolution (in arcsec) compared to the highest resolution spectroscopic survey available, e.g. GALFA-HI DR2 (4’). The available inferometric gas data is also having higher resolution to the full-sky polarimetric survey like Planck (5’). With these high inferometric data available, our method can potentially provide a more clear physical picture than current polarimetric survey both in the direction of magnetic field
using VGT and the Mach numbers based on our work. 6. SUMMARY
In this paper, we analyze the statistics of gradient amplitudes through simple statistics and the concept of filaments and find their correlation to the Mach numbers. To summarize: 1. Both intensity and centroid gradients are filamentary and the filaments’ width and orientations are correlated with Ms and MA 2. We found a nice linear relationship between the dispersion of amplitudes of gradients and Ms 3. In the framework of gradient amplitude filaments (a) We found a clear linear law on the normalized standard deviation σ/µ for both gradient amplitudes of velocity and intensities with respect to Ms (b) Similarly, the orientation of filaments are highly correlated to both Ms and MA . In particular, there is a strong linear dependence for the alignments of filaments to MA . 4. We applied our method to observation and find out that in this piece of GALFA-HI region the high-latitude part has a significantly higher sonic Mach number compared to those close to the galactic plane. This region is also lower in Alfvenic Mach number compared to the low-latitude region. Acknowledgments. We acknowledge Bo Yang and Junda Chen for their careful reading. We acknowledge the support the NSF grant DMS 1622353 and AST 1715754.
REFERENCES Andersson, B.-G., Lazarian, A., & Vaillancourt, J. E. 2015, ARA&A, 53, 501 Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, The Astrophysical Journal, 443, 209 Beresnyak, A., Lazarian, A., & Cho, J. 2005, ApJL, 624, L93 Beck, R. 2015, Magnetic Fields in Diffuse Media, 407, 507 Brandenburg, A., & Lazarian, A. 2013, SSRv, 178, 163 Burkhart, B., & Lazarian, A. 2012, ApJL, 755, L19 Burkhart, B., Lazarian, A., & Gaensler, B. M. 2012, The Astrophysical Journal, Volume 749, Issue 2, article id. 145, 16 pp. (2012)., 749 Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113 Chepurnov, A., & Lazarian, A. 2010, The Astrophysical Journal, Volume 710, Issue 1, pp. 853-858 (2010)., 710, 853 Cho, J., & Lazarian, A. 2002, Physical Review Letters, vol. 88, Issue 24, id. 245001, 88 —. 2003, Monthly Notices of the Royal Astronomical Society, Volume 345, Issue 12, pp. 325-339., 345, 325
2 While the spectroscopic gas data often take the same price as that in BLASTPol, publicly available data 10 years ago, e.g. Herschel, COMPLETE, already allows us to obtain magnetic field tracing with comparable resolution to PLANCK.
Cho, J., Lazarian, A., & Vishniac, E. 2001, The Astrophysical Journal, Volume 564, Issue 1, pp. 291-301., 564, 291 Clark, S. E., Peek, J. E. G., & Putman, M. E. 2014, ApJ, 789, 82 Clark, S. E., Hill, J. C., Peek, J. E. G., Putman, M. E., & Babler, B. L. 2015, Physical Review Letters, 115, 241302 Correia, C., Lazarian, A., Burkhart, B., Pogosyan, D., & De Medeiros, J. R. 2016, ApJ, 818, 118 Crutcher, R. M., Hakobian, N., & Troland, T. H. 2010, MNRAS, 402, L64 Dolginov, A. Z., & Mitrofanov, I. G. 1976, Ap&SS, 43, 291 Draine, B. T. 2009, Cosmic Dust - Near and Far, 414, 453 Draine, B. T. 2011, Physics of the interstellar and intergalactic medium (Princeton University Press), 540 Esquivel, A., & Lazarian, A. 2005, ApJ, 631, 320 Falceta-Gonc¸alves, D., Lazarian, A., & Kowal, G. 2008, ApJ, 679, 537-551 Federrath, C., Roman-Duval, J., Klessen, R. S., Schmidt, W., & Mac Low, M.-M. 2010, A&A, 512, A81 Gaensler, B. M., Haverkorn, M., Burkhart, B., et al. 2011, Nature, Volume 478, Issue 7368, pp. 214-217 (2011)., 478, 214 Goldreich, P. ;Sridhar, S. 1995, The Astronomical Journal, 438, 763 Gonz´alez-Casanova, D. F., & Lazarian, A. 2017, ApJ, 835, 41 Gonz´alez-Casanova, D. F., Lazarian, A., & Burkhart, B. 2017, arXiv:1703.03035 Gonz´alez-Casanova, D. F., & Lazarian, A. 2016 Jokipii, J. R. 1966, ApJ, 146, 480
7 Kandel, D., Lazarian, A., & Pogosyan, D. 2016, MNRAS, 461, 1227 Kandel, D., Lazarian, A., & Pogosyan, D. 2017, MNRAS, 464, 3617 Kowal, G., Lazarian, A., Vishniac, E. T., & Otmianowska-Mazur, K. 2009, ApJ, 700, 63 Lazarian, A., Pogosyan, D., & Esquivel, A. 2002, Seeing Through the Dust: The Detection of HI and the Exploration of the ISM in Galaxies, 276, 182 Lazarian, a. 2006, 4 Lazarian, A. 2007, JQSRT, 106, 225 Lazarian, A., & Pogosyan, D. 2000, ApJ, 537, 720 Lazarian, A., & Pogosyan, D. 2004, ApJ, 616, 943 Lazarian, A., & Pogosyan, D. 2006, ApJ, 652, 1348 Lazarian, A., & Vishniac, E. T. 1999, The Astrophysical Journal, Volume 517, Issue 2, pp. 700-718., 517, 700
Lazarian, A., & Yuen, K. H. 2018, ApJ, 853, 96 Lazarian, A., Esquivel, A., & Crutcher, R. 2012, ApJ, 757, 154 Lazarian, A., Yuen, K. H., Lee, H., & Cho, J. 2017, arXiv:1701.07883 Lee, H., Lazarian, A., & Cho, J. 2016, ApJ, 831, 77 Mac Low, M.-M., & Klessen, R. S. 2004, Reviews of Modern Physics, 76, 125 McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565 Peek, J. E. G., Babler, B. L., Zheng, Y., et al. 2018, ApJS, 234, 2 Soler, J. D., & Hennebelle, P. 2017, A&A, 607, A2 Soler, J. D., Hennebelle, P., Martin, P. G., et al. 2013, ApJ, 774, 128 ApJ, 130, 241 Yan, H., & Lazarian, A. 2008, ApJ, 673, 942-953 Yuen, K. H., & Lazarian, A. 2017, ApJL, 837, L24 Yuen, K. H., & Lazarian, A. 2017, arXiv:1703.03026
