Modelling the Galactic magnetic field on the plane in two dimensions

We present a method for parametric modelling of the physical components of the Galaxy's magnetized interstellar medium, simulating the observables and mapping out the likelihood space using a Markov Chain Monte Carlo analysis. We then demonstrate it using total and polarized synchrotron emission data as well as rotation measures of extragalactic sources. With these three data sets, we define and study three components of the magnetic field: the large-scale coherent field, the small-scale isotropic random field and the ordered field. In this first paper, we use only data along the Galactic plane and test a simple two-dimensional (2D) logarithmic spiral model for the magnetic field that includes a compression and a shearing of the random component giving rise to an ordered component. We demonstrate with simulations that the method can indeed constrain multiple parameters yielding measures of, for example, the ratios of the magnetic field components. Though subject to uncertainties in thermal and cosmic ray electron densities and depending on our particular model parametrization, our preliminary analysis shows that the coherent component is a small fraction of the total magnetic field and an ordered component comparable in strength to the isotropic random component is required to explain the polarization fraction of synchrotron emission. We outline further work to extend this type of analysis to study the magnetic spiral arm structure, the details of the turbulence as well as the 3D structure of the magnetic field.