Stochastic excitation of waves in magnetic stars -- I. Scaling laws for the modes amplitudes

Stellar oscillations are key to unravelling stars' properties, such as their mass, radius and age. Amplitudes of acoustic modes in solar-like stars are intrinsically linked to their convective turbulent excitation source, which in turn is influenced by magnetism. In the observations of the Sun and stars, the amplitude of the modes is modulated following their magnetic activity cycles: the higher the magnetic field, the lower the modes' amplitudes. When the magnetic field is strong, it can even inhibit the acoustic modes, which are not detected in a majority of solar-like stars presenting a strong magnetic activity. Magnetic fields are known to freeze convection when stronger than a critical value: an "on-off" approach is used in the literature. In this work, we investigate the impact of magnetic fields on the stochastic excitation of acoustic modes. First, we generalise the forced wave equation formalism, including the effects of magnetic fields. Second, we assess how convection is affected by magnetic fields using results from Magnetic Mixing-Length Theory. We provide the source terms of stochastic excitation, including a new magnetic source term and the Reynolds stresses. We provide scaling laws for the amplitudes of the modes, taking into account both the driving and the damping. Those scalings are based on the inverse Alfv\'en dimensionless parameter: the damping increases with the magnetic field and reaches a saturation threshold when the magnetic field is strong. The driving of the modes diminishes when the magnetic field becomes stronger, the turbulent convection being weaker. As expected from the observations, we find that a higher magnetic field diminishes the resulting modes amplitudes. Evaluating the inverse Alfv\'en number in stellar models provides a means to estimate the expected amplitudes of acoustic modes in magnetic active solar-type stars.