Astro 201 – Radiative Processes – Problem Set 11

Due in class.

Readings: RL 8.1, 8.2; however much of RL 3.1–3.3 and Blandford’s notes that you like

Problem 1. Typical vs. Average (or Lies, Damn Lies, and Statistics)

Consider a non-relativistic electron moving at speed v ≪ c. A single photon of original energy ϵ Compton scatters off the electron and suffers an energy shift of Δϵ. Work in the same Thomson limit that we have been working in throughout lecture.

(a) Estimate a typical value for the fractional change in photon energy, Δϵ/ϵ. Express your answer symbolically. By “typical,” we mean any old photon that comes in from any old direction.

(b) Consider this same electron flying through a sea of photons all of energy ϵ. Many photons get inverse Compton scattered to and fro.

Averaged over all the photons that got scattered, what is the mean fractional change in photon energy, ⟨Δϵ/ϵ⟩? You may use our expression for the inverse Compton power scattered by a single electron if you wish. Neglect the change in the electron’s velocity as it is bombarded by photons.

(c) Explain physically why parts (a) and (b) give different answers. (And then read Blandford page 175, if you wish. It is better to think about this problem and achieve your own understanding than to head straight for Blandford right away).

Problem 2. Faraday Rotation

Consider the propagation of light through a magnetized plasma. The magnetic field is uniform B⃗_{0} = B_{0}ẑ. Light travels parallel to ẑ.

An electron in the plasma feels a force from the electromagnetic wave, and a force from the externally imposed B-field. Its equation of motion reads

m˙v⃗ = −eE⃗ − \frac{e}{c} v⃗ × B⃗_{0}  (1)

where the electric field E⃗ can be decomposed into right-circularly-polarized (RCP) and left-circularly-polarized (LCP) waves:

E⃗ = E_{0}(x̂ ∓ iŷ)e^{i(k_{∓}z−ωt)}  (2)

where it is understood that the real part should be taken. The upper sign (-) corresponds to RCP waves, while the lower sign (+) corresponds to LCP waves.

In the equation of motion above, we have neglected the Lorentz force from the wave’s B-field, since it is small (by v/c) compared to the force from the wave’s E-field.

(a) Prove that the solution of the equation of motion reads

v⃗ = \frac{−ie}{m(ω ± ω_{cyc})}E⃗  (3)

where ω_{cyc} = eB_{0}/mc. This is the hardest part of the derivation for the dispersion relation of RCP and LCP waves. But it is fairly straightforward.

(b) Rybicki & Lightman Problem 8.3

Problem 3. Multiple Multipoles

(a) Write down the electric field a distance r away from a monopole of charge q.

(b) Someone moves another monopole of charge −q next to the original monopole. The two charges are separated by a distance b. Derive, to order-of-magnitude, the factor by which this dipole electric field is reduced from the monopole field.

(c) Someone moves two more charges, −q and q, into position to form a square. The edge of the square has length b. Going around the square, the charges are −q, q, −q, and q. Derive, to order-of-magnitude, the factor by which this quadrupole electric field is reduced from the dipole field.

(d) Draw a picture of a pure electric octopole, that is, a charge distribution for which the electric field decreases as 1/r^{5} (and no less gradually). For bonus points, draw a picture of an electric hexadecapole (electric field dies no less gradually than 1/r^{6}).

(e) Now imagine the dipole and quadrupole configurations in (b) and (c) rotating about their centers-of-charge with frequency ν. We have a rotating barbell and a rotating square, respectively. (Think CO and H_{2}).

To order-of-magnitude, what is the maximum distance from each object inside of which the electric fields are nearly perfectly in phase with the rotation?

This is the boundary of the near zone, inside of which the electric field geometry rotates with frequency ν as if it were a rigid body.

(f) Electromagnetic waves are emitted at the boundary of the near zone, into the far (radiation) zone.

Derive (one line of argument suffices) the factor by which the power carried by waves emitted by the rotating quadrupole is smaller than the power carried by waves emitted by the rotating dipole.