Turbulence Spectra from Doppler-shifted Spectral Lines

Turbulence is a key element of the dynamics of astrophysical fluids, including those of interstellar medium, clusters of galaxies and circumstellar regions. Turbulent motions induce Doppler shifts of observable emission and absorption lines. In the review we discuss new techniques that relate the spectra of underlying velocity turbulence and spectra of Doppler-shifted lines. In particular, the Velocity-Channel Analysis (VCA) makes use of the channel maps, while the Velocity Coordinate Spectrum (VCS) utilizes the fluctuations measured along the velocity axis of the Position-Position Velocity (PPV) data cubes. Both techniques have solid foundations based on analytical calculations as well as on numerical testings. Among the two the VCS, which has been developed quite recently, has two advantages. First of all, it is applicable to turbulent volumes that are not spatially resolved. Second, it can be used with absorption lines that do not provide good spatial sampling of different lags over the image of turbulent object. In fact, numerical testing shows that measurements of Doppler shifted absorption lines over a few directions is sufficient for a reliable recovering of the underlying spectrum of the turbulence. Our comparison of the VCA and the VCS with a more traditional technique of Velocity Centroids, shows that the former two techniques recover reliably the spectra of supersonic turbulence, while the Velocity Centroids may have advantages for studying subsonic turbulence. In parallel with theoretical and numerical work on the VCA and the VCS, the techniques have been applied to spectroscopic observations. We discuss results on astrophysical turbulence obtained with the VCA and the VCS. WHAT CAN TURBULENT SPECTRA TELL US? As a rule astrophysical fluids are turbulent and the turbulence is magnetized. This ubiquitous turbulence determines the transport properties of interstellar medium (see Elmegreen & Falgarone 1996, Stutzki 2001, Balesteros-Peredes et al. 2006) and intracluster medium (see Sunyaev, Norman & Bryan 2003, Ensslin & Vogt 2006, Lazarian 2006), many properties of Solar and stellar winds (see Hartman & McGregor 1980) etc. One may say that to understand heat conduction, propagation of cosmic rays and electromagnetic radiation in different astrophysical environments it is absolutely essential to understand the properties of underlying magnetized turbulence. The fascinating processes of star formation (see McKee & Tan 2002, Elmegreen 2002, Mac Low & Klessen 2004) and interstellar chemistry ( see Falgarone et al. 2006 and references therein) are also intimately related to properties of magnetized compressible turbulence (see reviews by Elmegreen & Scalo 2004). From the point of view of fluid mechanics astrophysical turbulence is characterized by huge Reynolds numbers, Re, which is the inverse ratio of the eddy turnover time of a parcel of gas to the time required for viscous forces to slow it appreciably. For Re ≫ 100 we expect gas to be turbulent and this is exactly what we observe in HI (for HI Re ∼ 108). In fact, very high astrophysical Re and its magnetic counterpart magnetic Reynolds number Rm (that can be as high as Rm ∼ 1016) present a big problem for numerical simulations that cannot possibly get even close to the astrophysicallymotivated numbers. The currently available 3D simulations can have Re and Rm up to ∼ 104. Both scale as the size of the box to the first power, while the computational effort increases as the fourth power (3 coordinates + time), so the brute force approach cannot begin to resolve the controversies related to ISM turbulence. This caused serious concerns that while present codes can produce simulations that resemble observations, whether numerical simulations reproduce reality well (see McKee 1999, Shu et al. 2004). We believe that these concerns may be addressed via observational studies of astrophysical turbulence. Statistical description is a nearly indispensable strategy when dealing with turbulence. The big advantage of statistical techniques is that they extract underlying regularities of the flow and reject incidental details. Kolmogorov description of unmagnetized incompressible turbulence is a statistical one. For instance it predicts that the difference in velocities at different points in turbulent fluid increases on average with the separation between points as a cubic root of the separation, i.e. |δv| ∼ l1/3. In terms of direction-averaged energy spectrum this gives the famous Kolmogorov scaling E(k) ∼ 4πk2P(k) ∼ k5/3, where P(k) is a 3D energy spectrum defined as the Fourier transform of the correlation function of velocity fluctuations ξ (r) = 〈δv(x)δv(x + r)〉. Note that in this paper we use 〈...〉 to denote averaging procedure. The example above shows the advantages of the statistical approach to turbulence. For instance, the energy spectrum E(k)dk characterizes how much energy resides at the interval of scales k,k + dk. At large scales l which correspond to small wavenumbers k ( i.e. l ∼ 1/k) one expects to observe features reflecting energy injection. At small scales one should see the scales corresponding to sinks of energy. In general, the shape of the spectrum is determined by a complex process of non-linear energy transfer and dissipation. In view of the above it is not surprising that attempts to obtain spectra of interstellar turbulence have been numerous since 1950s (see Munch 1958). However, various directions of research achieved various degree of success (see Armstrong, Rickett & Spangler 1995). For instance, studies of turbulence statistics of ionized media were more successful (see Spangler & Gwinn 1990) and provided the information of the statistics of plasma density at scales 108-1015 cm. However, these sort of measurements provide only the density statistics, which is an indirect measure of turbulence. Velocity statistics is much more coveted turbulence measure. Although, it is clear that Doppler broadened lines are affected by turbulence, recovering of velocity statistics was extremely challenging without an adequate theoretical insight. Indeed, both velocity and density contribute to fluctuations of the intensity in the Position-Position-Velocity (PPV) space. In what follows we discuss how the observable Doppler-shifted lines can be used to recover a spectrum of turbulent velocity using two new techniques, that, unlike other mostly empirical techniques, have solid theoretical foundations. How to obtain using spectroscopic observations other characteristics of turbulence, e.g. higher order statistics, anisotropies has been reviewed earlier (see Lazarian 2004). FIGURE 1. Left Panel: PPV data cube. Illustration of the concepts of the thick and thin velocity slices. the slices are thin for the PPV images of the large eddies, but thick for the images of small eddies. Right Panel: Illustration of the VCS technique. For a given instrument resolution large eddies are in the high resolution limit, while small eddies are in the low resolution limit. VCA AND VCS: TWO WAYS TO ANALYZE SPECTRAL DATA Spatial spectra obtained by taking Fourier transform of channel maps had been widely used to study HI before we conducted our theoretical study of what those spectra mean in Lazarian & Pogosyan (2000, henceforth LP00). The channel maps correspond to the velocity slices of PPV cubes as shown in Figure 1 and one may naturally ask a question whether anything depends on the thickness of the channel. It is intuitively clear that if medium is optically thin and the velocity is integrated over the whole spectral line, the fluctuations can depend only on density inhomogeneities. It is also suggestive that the contribution of the velocity fluctuations may depend on whether the images of the eddies under study fit within a velocity slice or their velocity extend is larger than the slice thickness (see Figure 1). According to LP00 this results in the assymptotics that correspond to “thin” and “thick” velocity channels. Note, that these questions were not posed by the earlier research. This resulted in channel map spectra with different spectral indexes the relation of which to the underlying velocity fluctuations was unclear (see Green 1990 and references therein). Some spectral data, for instance, optically thick CO data were traditionally analyzed in differently (see Falgarone & Puget 1995), namely, spectra of total intensities was studied. The origin of this spectrum and its relation to the underlying velocity and density fluctuations was established in Lazarian & Pogosyan (2004, henceforth LP04). This work also clarified the effects of absorption that were reported for HI data. In terms of the techniques of turbulence study LP04 deals with the VCA, but in the case when absorption is present. A radically different way of analyzing spectroscopic data is presented in Lazarian & Pogosyan (2006, henceforth LP06) and Chepurnov & Lazarian (2006a). There the spectra along the V-axis of the PPV cube are studied (see Figure 2). The technique was termed Velocity Coordinate Spectrum (VCS) in Lazarian (2004). The formalism can be traced to LP00 work, where the expressions that relate the spectrum of fluctuations along the velocity coordinate and the underlying velocity spectrum were obtained. However, it took some time to understand the advantages that the VCS provides for the practical handling of the emission and absorption data. The first analysis of the data using the VCS is performed by Chepurnov & Lazarian (in preparation). Numerical testing of the technique is provided in Chepurnov & Lazarian (2006b). BASICS OF THE FORMALISM Below we provide a brief introduction to the mathematical foundations of the VCA and the VCS (see more in LP00, LP04 and LP06). Our goal is to relate the statistics that can be obtained through spectral line observations, for instance, the structure function of the intensity of emission IX(v)