Introduction to Astrophysics II, 29:120
Winter 2009
First Homework Set... January 22, 2009. Due January 29, 2009

Show calculations and give reasons for your answers. Don't go around confused and despondent; if you do not know how to get started, ask me for help. The purpose of problem sets is to promote thinking and lead to understanding, not produce a confiteor of revealed truth.

(1) A hydrogen plasma has a number density of $5 \times 10^6$ atoms/m$^3$, and there is a magnetic field $B_0=5 \times 10^{-9}$ Tesla. What is the Alfvén speed in this plasma?

(2) Show by direct substitution that waves are solutions to the equation derived in class,

$$\frac{\partial^2 b}{\partial t^2} - V_A^2 \frac{\partial^2 b}{\partial z^2} = 0$$ (1)

(3) (a) Write down the wave function $b(z,t)$ corresponding to the magnetic field in an Alfvén Wave. (b) Using intermediate equations derive in class, derive an expression for the velocity wave function $v_x(z,t)$.

(4) An Alfvén wave is propagating in the plasma described in problem # 1. It has an amplitude of $\vert b_x\vert = 0.1 B_0$ (as defined in lecture) and a wavelength of $\lambda = 7 \times 10^5$ km. What is the maximum acceleration of the plasma due to the presence of this wave?

(5) If the Alfvén wave described in Problem # 4 were propagating in a single direction, what would the associated energy flux be? Hint: The energy flux $S$ is given by $S= \rho_E V$ where $\rho_E$ is the energy density and $V$ is the speed at which the energy is transported.

(6) Consider an imaginary spherical surface of radius $R_0$ in a moving, constant density fluid. Measurements show that the fluid velocity on the surface of the sphere is given by

$$\vec{v}(r=R_0,\theta,\phi) = (V_0 \cos \theta + V_1 \sin \theta ) \hat{e}_r - V_0 \sin \theta \hat{e}_{\theta}$$ (2)

Explain why this information tells you that there is a source of fluid interior to the sphere. Derive an expression for the rate at which mass is being added to the fluid inside the sphere. Hint: Go back and look at the equation of continuity from the equations of hydrodynamics, and think about Gauss's theorem from electricity and magnetism.

(7) Problem 12.1

(8) Problem 12.2