March 2, 2011 Fill in derivation from last lecture Polarization of Thomson Scattering No class Friday, March 11

If Show that Need two identities: So… Now

Magnitudes of E(rad) and B(rad): Poynting vector is in n direction with magnitude

The Dipole Approximation Generally, we will want to derive for a collection of particles with You could just add the ‘s given by the formulae derived previously, but then you would have to keep track of all the t retard (i) and R retard (i) The Dipole Approximation

One can treat, however, a system of size L with “time scale for changes” tau where so differences between t ret (i) within the system are negligible Note: since frequency of radiation Ifthen or This will be true whenever the size of the system is small compared to the wavelength of the radiation. The Dipole Approximation

Thomson Scattering Rybicki & Lightman, Section 3.4

Thomson Scattering EM wave scatters off a free charge. Assume non-relativistic: v<<c. E field electron e = charge Incoming E field in direction Incoming wave: assume linearly polarized. Makes charge oscillate. Wave exerts force : r = position of charge

Energy per second, Cross-sections: time averaged power / solid angle The total power is obtained by integrating over all solid angle: Cross section / solid angle, Polarized incoming light Total cross-section, Integrated over solid angle

Electron Scattering for un-polarized radiation Unpolarized beam = superposition of 2 linearly polarized beams with perpendicular axes

Differential Cross-section Average for 2 components Thomson cross-section for unpolarized light

NOTES: is independent of frequency of incoming wave Total cross-section is same as unpolarized case. Forward-Backward symmetry incident

Polarization of scattered radiation The scattered radiation is polarized, even if the incident radiation is unpolarized.

Incident light linear polarized in direction #1 Incident light linear polarized in direction #2 Unpolarized incident light No light along dipole axis

Degree of linear polarization: Degree of linear polarization polarized intensity total intensity generally, get net polarization incident wave, unpolarized Π=100% complete polarization Π=0 No net polarization in forward direction

Summary of Thomson Scattering (1) So electrons are much more effective scatterers than protons  by factor of 4x10^6 (2) forward and backward scattering identical (3) σ(pol) = σ(unpolar) in our classical treatment of the electron, it has no preferred direction (i.e. no spin)

(4) Scattered E-field is polarized = percent polarization (5) Thomson scattering σ T is independent of frequency (“grey”)

R&L Problem 3.2: Cyclotron Radiation A particle of mass m, charge e, moves in a circle of radius a at speed V ┴ << c. Define x-y-z coordinate system such that n is in the y-z plane (a) What is the power emitted per unit solid angle in a direction n, at angle θ to the axis of the circle?

(a) What is the power emitted per unit solid angle in a direction n, at angle θ to the axis of the circle? Consider a point at distance r from the origin magnitude of poynting vector radiation part of E field so power/solid angle

What is E rad ? Define unit vectors normal to the plane containing n, i.e. y-z plane Particle has speed V ┴ in circle of radius a, so angular velocity of particle position of particle at time t velocity of particle

acceleration of particle at time t Now unit vectors are related by So

Power / steradian as a function of t The time-average power/steradian since

(b) What is the polarization of the radiation as a function of θ ? Recall the discussion of Stokes parameters: we write E in terms of x- and y- components Identify x-component with y-component with then

and the Stokes parameters are Where we have let So (1) the radiation is 100% elliptically polarized (2) principle axes of polarization ellipse are (2) At θ=0 left-hand circular polarization θ=π/2 linear polarization along θ=π right-hand circular polarization and

(c) What is the spectrum of the radiation? Only cos ωt and sin ωt terms  spectrum is monochromatic at frequency ω

(d) Suppose the particle is moving in a constant magnetic field, B What is ω, and total power P? F B Lorentz force is balanced by centripetal force Lorentz force So gyro frequency of particle in B field

from part (a) where

(e) What is the differential and total cross-section for Thomson scattering of circularly polarized radiation? Equate the electric part of the Lorentz force = eE with centripetal force = mrω 2 and use our expression for for a circularly moving charge

Then Recall So, differential cross section Total cross-section Thomson cross-section

Rybicki & Lightman Problem 3.4 Consider an optically thin cloud surrounding a luminous source. The cloud consists of ionized gas. Assume that Thomson scattering is the only important source of optical depth, and that the luminous source emits unpolarized radiation. (a) If the cloud is unresolved, what polarization is observed? If the angular size of the cloud is smaller than the angular resolution of the detector, the polarization of the different parts of the cloud cancel  no net polarization R=1pc

(b) If the cloud is resolved, what is the direction of the polarization as a function of position on the sky? Assume only “single-scattering” – i.e. each photon scatters only once. θ At each θ, the incident, unpolarized wave can be decomposed into 2 linearly polarized waves: one in the plane of the paper, one normal to the plane of the paper. These scatter into new waves in ratio cos 2 θ : 1 Thus, the component normal to the paper dominates the other Integrating over every θ along a given line of sight results in net polarization which is normal to radial line:

Net result:

(c) If the central object is clearly seen, what is an upper bound for the electron density of the cloud, assuming that the cloud is homogeneous? To see the central object,