Multipolar Magnetic Elds in Rotating Ap Stars: Modeling of Observable Quantities

Magnetic eld is present in most of the chemically peculiar stars of the upper main sequence. For a long time, the component of the magnetic eld along the line of sight and the magnetic eld 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 elds of such a class of stars. On the other hand, the analytical modeling techniques, that is, the capability to reproduce the observable quantities { predicted by a given magnetic connguration { through simple analytical formu-lae, is still limited only to some of the quantities nowadays measurable and for the simplest magnetic conngurations. This work gives a contribution in order to ll this gap. Through the formalism of the spherical tensorial calculus , analytical formulae for most of the observable quantities are provided for a magnetic eld of arbitrary complexity. Namely, the magnetic eld 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 eld components which represent the observable quantities. The mean longitudinal magnetic eld, the so{called mean asymmetry of the longitudinal magnetic eld and the mean quadratic magnetic eld, 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 connguration. These sums can numerically be performed, or they can be further developed in order to provide simpler analytical expressions in the case of a deened multipolar expansion: in partic-Send ooprint requests to: S. Bagnulo ular, such simpler analytical expressions are given in the case of a magnetic eld due to the superposition of a dipole with a quadrupole. Finally, few diagrams of the observable quantities predicted by such a connguration are shown.