Introduction to Astrophysics II, 29:120
Winter, 2006-2007
Fourth Homework Set...February 12, 2007. Due February 16, 2007;5PM

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 atoms/cm$^3$, and there is a magnetic field $B_0=5 \times 10^{-5}$ G. What is the Alfvén speed in this plasma?

(2) 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?

(3) If the Alfvén wave described in Problem # 2 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.

(4) We see the solar corona mainly because photospheric light is Thomson scattered out of its original direction of propagation and into our line of sight. An empirical estimate of the electron density in the solar corona is

$$n_e(r) = 1.3 \times 10^5 \left[ \frac{1}{x^2} + \frac{25}{x^4} + \frac{300}{x^8} \right] \mbox{ cm}^{-3}$$ (1)

where $x \equiv \frac{r}{R_{\odot}}$. Calculate the amount by which the intensity of photospheric light (the light you see as the disk of the Sun) is diminished by Thomson scattering in the corona.

(5) 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.

(6) Assume that a velocity field $\vec{v}$ is given as the gradient of a scalar potential,

$$\vec{v} = -\nabla \phi$$ (3)

in which the potential is a solution of Laplace's equation. Assuming that the density in the fluid is initially uniform, what can you say about the time development of the density. Say it in equations!

(7) Problem 11.11

(8) Problem 11.12