## 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.