4. At ISM densities, the mean free path for particle-particle collisions is very large. In collisionless shocks, energy can instead be dissipated on much shorter scales by means of collective interactions between charged particles. a. Show that, for free electrons and protons in a population with kinetic temperature T, the cross section for mutual collisions is σ ~ лe/(k2T2). In the absence of other scattering processes, what is the mean free path traversed by such a particle in a T~104 K gas with density n = 1 cm-³? Hint: A charged particle will be significantly deflected by another charged particle at a radius where their joint electrostatic potential energy roughly equals their kinetic energy. b. A frequency associated with collective electromagnetic phenomena, involving the fields produced by populations of charged particles, is the plasma frequency, p = √4лene/me, where ne is the free-electron density, and me is the electron mass. Derive wp by considering an overall neutral gas of free electrons and protons, which is initially polarized (e.g., by applying an external electric field) such that, say, all the electrons are shifted in parallel by a distance x from their equilibrium positions. The external field is then turned off. Write the equation of motion of an electron under the influence of the electric field produced by all the other charges, and show that all the electrons oscillate collectively in harmonic motion, with frequency wp. Find the collisionless interaction length, l = vo/wp, for a shock velocity vo 104 km sand an ISM density n = 1 cm³. ~ Hint: In the equation of motion, the restoring force on an electron is its charge times the electric field from all the other displaced electrons. That electric field is just like the field within a parallel-plate capacitor, E = 4лoe, with σ, the surface density of electric charge.