Multipolar magnetic fields in rotating Ap stars: modeling of observable quantities

S. Bagnulo, M Landi Degl'Innocenti. and E Landi Degl'Innocenti

Magnetic field is present in most of the chemically peculiar stars of the upper main sequence. For a long time, the component of the magnetic field along the line of sight and the magnetic field modulus, averaged on the stellar disk, were the only diagnostic contents of the observational techniques. Recent developments in the application of the theory of line formation, joined to the increased sensitivity of instrumental techniques, augmented the number of observable quantities related to the magnetic fields of such a class of stars. On the other hand, the analytical modelling techniques, that is, the capability to reproduce the observable quantities - predicted by a given magnetic configuration - through simple analytical formulae, is still limited only to some of the quantities nowadays measurable and for the simplest magnetic configurations. This work gives a contribution in order to fill this gap.

Through the formalism of the spherical tensorial calculus, analythical formulae for most of the observable quantities are provided for a magnetic field of arbitrary complexity.

Namely, the magnetic field is thought of as originated by a multipolar expansion of arbitrary order (dipole, plus quadrupole, plus octupole,...) and it is expressed through spherical tensors and spherical harmonics. By using such a formalism, it is straightforward to carry on the integration over the stellar disk of the particular combinations of the magnetic field components which represent the observable quantities. The mean longitudinal magnetic field, the so-called mean asymmetry of the longitudinal magnetic field and the mean quadratic magnetic field, together with the quantities related to the observed broadband linear polarization, are expressed through suitable sums of terms including the spherical components of tensors describing the magnetic configuration. These sums can numerically be performed, or they can be further developed in order to provide simpler analytical expressions in the case of a defined multipolar expansion: in particular, such simpler analytical expressions are given in the case of magnetic field due to the superposition of a dipole with a quadrupole. Finally, few diagrams of the observable quantities predicted by such a configuration are shown.