An exact analytical solution for the weakly magnetized flow around an axially symmetric paraboloid, with application to magnetosphere models

TL;DR

This paper presents an exact analytical solution for the flow and magnetic field around a paraboloid, applicable to astrophysical magnetosphere modeling, incorporating incompressible and mildly compressible flow conditions.

Contribution

It introduces a novel analytical method using Euler potentials and Cauchy's integral to model magnetic fields in flow around paraboloids, extending to compressible flows.

Findings

Derived an exact solution for potential flow around a paraboloid.

Extended the model to include magnetic field advection in ideal MHD.

Generalized the flow to mildly compressible conditions using Bernoulli's principle.

Abstract

Rotationally symmetric bodies with longitudinal cross sections of parabolic shape are frequently used to model astrophysical objects, such as magnetospheres and other blunt objects, immersed in interplanetary or interstellar gas or plasma flows. We discuss a simple formula for the potential flow of an incompressible fluid around an elliptic paraboloid whose axis of symmetry coincides with the direction of incoming flow. Prescribing this flow, we derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics for the case of an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. Our solution procedure employs Euler potentials and Cauchy's integral formalism based on the flow's stream function and isochrones. Furthermore, we use a particular renormalization procedure that allows us to generate more general analytical expressions modeling the deformations experienced by arbitrary scalar or vector-valued fields embedded in the flow as they are advected first toward and then past the parabolic obstacle. Finally, both the velocity field and the magnetic field embedded therein are generalized from incompressible to mildly compressible flow, where the associated density distribution is found from Bernoulli's principle.