Radius constraints and minimal equipartition energy of relativistically moving synchrotron sources

A measurement of the synchrotron self-absorption flux and frequency provides tight constraints on the physical size of the source and a robust lower limit on its energy. This lower limit is also a good estimate of the magnetic field and electrons' energy, if the two components are at equipartition. This well-known method was used for decades to study numerous astrophysical sources moving at non-relativistic (Newtonian) speeds. Here we generalize the Newtonian equipartition theory to sources moving at relativistic speeds including the effect of deviation from spherical symmetry expected in such sources. Like in the Newtonian case, minimization of the energy provides an excellent estimate of the emission radius and yields a useful lower limit on the energy. We find that the application of the Newtonian formalism to a relativistic source would yield a smaller emission radius, and would generally yield a larger lower limit on the energy (within the observed region). For sources where the Synchrotron-self-Compton component can be identified, the minimization of the total energy is not necessary and we present an unambiguous solution for the parameters of the system.