The Scaling of Coronal Models from One Star to Another

The requirements that must be met in order that stationary numerical corona models can be scaled from one star to another are discussed. A corona model is a solution of the conservation equations for mass, momentum, and energy, subject to appropriate boundary conditions, and of the equation of state. In general, the mass M and radius R of the star enter these equations and boundary conditions as free parameters. A given solution can be scaled to other stars only if all equations can be rewritten in such a form that M and R do no longer appear explicitly as free parameters, but only implicitly as scaling factors of the variables. An adequate means to find these scaling factors is a homologous transformation: one multiplies all variables and parameters by separate constants (i.e., scaling factors) and requires that the equations and boundary conditions remain valid. This leads to a set of nonlinear relations between the transformation constants. Only if in this set the two constants associated with M and R can be chosen independently, can a given numerical corona model be scaled to arbitrary stars.