Obtaining magnetic field strength using differential measure approach and velocity channel maps

We introduce two new ways of obtaining the strength of plane-of-sky (POS) magnetic field by simultaneous use of spectroscopic Doppler-shifted lines and the information on magnetic field direction. The latter can be obtained either through polarization measurements or using the velocity gradient technique. We show the advantages that our techniques have compared to the traditional Davis-Chandrasekhar-Fermi (DCF) technique of estimating magnetic field strength from observations. The first technique that we describe in detail employs structure functions of velocity centroids and structure functions of Stokes parameters. We provide analytical expressions for obtaining magnetic field strength from observational data. We successfully test our results using synthetic observations obtained with results of MHD turbulence simulations. We measure velocity and magnetic field fluctuations at small scales using two, three and four point structure functions and compare the performance of these tools. We show that, unlike the DCF, our technique is capable of providing the detailed distribution of POS magnetic field and it can measure magnetic field strength in the presence of both velocity and magnetic field distortions arising from external shear and self-gravity. The second technique applies the velocity gradient technique to velocity channel maps in order to obtain the Alfven Mach number and uses the amplitudes of the gradients to obtain the sonic Mach number. The ratio of these two Mach numbers provides the intensity of magnetic field in the region contributing to the emission in the channel map. We test the technique and discuss obtaining the 3D distribution of POS galactic Magnetic field with it. We discuss the application of the second technique to synchrotron data.