Problems

=School of modern astrophysics (SOMA - 2014)=

Problems
__Main list__ 1. **Group polar.** Draw a group polar (i.e., the polar diagram of the group velocity dependence on the angle between magnetic field and the group velocity) for V A near c s. Hint: Use the dispersion relations ω=ω(k) for the ideal MHD modes (Alfvenic, fast and slow magnetosonic) giving the expression for the group velocity. 2. **MRI (magneto-rotational instability).** Obtain the MRI dispersion relation (ω 2 =ω 2 (k) ) and show that the instability does exist. Hint: Consider the Alfvenic mode only (the density remains constant). In the zero approximation there is Keplerian rotation, and magnetic field is perpendicular to the disk plane (B z only). Disturbances of the velocity and magnetic field have the azimuthal components only, the wave vector is in the z direction.

__Additional list__ 1. How to distinguish the gravitational red shift and the proper motion for an individual White Dwarf? 2. Model the cylindrical Sun spot within the force-free approximation. Hint: solve the Laplace equation in the cylindrical geometry, using only one cylindrical basis function for simplicity.