Eline Tolstoyhttp://www.astro.rug.nl/~etolstoy/astroa07/

3. Basic Theory ofRadiation Fields

Radiative Processesin Astrophysics

Lorentz Force Lawdescription of radiation in terms of electromagnetic theory

in the non-relativistic limit, the Lorentz force exerted on a particle with chargeq, of velocity, v in an Electric field, E and magnetic Field B:

Lorentz Force

the force due to the magnetic field is always perpendicular to both the directionof the velocity vector and the field.

e.g., providing the E field is negligible, charged particles are forced to spiralaround magnetic field lines and cannot cross them except by collisions. Thismeans that closed magnetic field lines tend to trap charged particles.

Lorentz Force (contd)

so the rate of work done by fields on a particle:

v!(v"B) =0 because B worksperpendicular to motion.

for non-relativisticparticles

generalizing to total force on a volume element containing many charges,the force per unit volume is:

the rate of work done by the field per unit volume

the rate of work per unit volume is and is equal to the rate ofincrease of mechanical energy density.

Review of Maxwell’s Equationselegant and concise way to state the fundamentals of electricity and magnetism

describes thebehaviour of bothelectric andmagnetic fields aswell as theirinteractions withmatter

Poynting vectorD=#EB=µH

using Maxwell’s Eqns, consider the work done per unit volume on a particledistribution (dotting Ampere’s law):

and Faraday’s law

using

Poynting (contd)

Poynting vectorrate of change of total energy density

The Poynting flux has some peculiar features, namely - it appears to say thata charged bar magnet has an energy flux that circulates around the bar in atoroidal sense, which is not very meaningful, however if one integrates theequation over some finite volume then the rhs can be converted into an integralover the surface volume element of Poynting flux and this is well defined andfree of peculiarities

Poynting (contd)

The rate of change of total energy within volume,V, is equal to the net inwardflow of energy through the bounding surface, $.

Integrating over volume element and using divergence theorem

Electrostatics: both E and B decrease like r-2. This implies S decreases like r-4

and thus the integral goes to zero since the surface area increases only as r2.

For time varying fields E and B decrease like r-1, and therefore the integral cancontribute a finite amount to the rate of change of energy of the system. Thisenergy flowing in (or out) at large distances is called RADIATION.

Wave EquationMaxwell’s Equations is a vacuum(%=0, j=0):

A basic feature of these eqns is the existence of traveling wave solutions thatcarry energy. Taking the curl of 3rd eqn & using 4th, :

Solving Wave EquationThe general solution of this wave equation,has the form:

Where â1 & â2 are unit vectors; E0 & B0 are complex constants and k=kn isthe wave vector and & is the frequency.

This solution represents waves traveling in the n direction. Bysuperposing such solutions propagating in all directions and with allfrequencies we can construct the most general solution of the source freeMaxwell’s Eqns.

The first 2 tell us that both â1 and â2 are transverse to the direction ofpropagation, k. With this can carry out cross products in second 2 eqns andsee that â1 and â2 are perpendicular to each other, ie., â1, â2 & k form a right-handed set.

Plane EM-wavesThe values of E0 and B0 are related:

thus can show that E0 = B0

and & = ck

The Radiation SpectrumFrom the TIME VARIATION OF THE ELECTRIC FIELD, andanalogously the magnetic field, follows the spectrum of the radiation.

The spectrum is the amount of energy per unit area per unit time per unitfrequency interval, and is most easily derived through a Fouriertransformation.

Consider a pulse of radiation that passes by an observer,

We only need to consider the E-field along one axis

The Fourier transform & it’s inverse are now defined:

Ê(w) contains the full frequency information of E(t)

The Radiation Spectrum (contd)Since the (average) amount of energy dW passing through a surfaceelement dA per unit time dt is given the time-averaged Poynting vector‹S› for the electric and magnetic parts:

The total energy per unit area in the pulse:

Spectral shapeThe fact that the time variation of the electrical field and its spectrum are related through a Fouriertransform makes it very convenient to derive a spectral shape from the characteristics of E(t)

em-pulse radiationspectra

a pulse of duration T has a spectrumstretching over a bandwidth of ~1/T

A periodic signal with frequency &0 fora duration, T will have a spectrumwidth 1/T centred on &0

A similar periodic signal with a decaytime of T (damped oscillator) will producea spectrum of bandwidth 1/T centered on&0, but without the higher and lowerfrequency wiggles of previous example

Polarisation

mono-chromatic plane waves,linearly polarised

Linearly polarised means the electric vector simply oscillates in directionâ1, which, with the propagation direction defines the plane of polarization.By superimposing solutions corresponding to two such oscillations inperpendicular we can construct the most general state of polarisation for awave of given k and &.

the vector E traces out an ellipse

Orthogonal components

Any (single frequency) E wave can be decomposed into 2 orthogonalwaves with amplitudes E1 and E2 and the same frequency (with differentphase). The resulting composite E traces out an ellipse.

Monochromatic polarisationThe vectors E1 and E2 can be written in amplitude/phase notation

In the laboratory frame (x, y) we can find the components of the field alongthe x and y axes^

^^

^

These vectors describe the tip of the electric field vector in the x-y plane.

The ellipse traced out by E in the lab frame, andalso in the frame aligned with the principle axes ofthe ellipse:

system rotated by angle ' with respect to lab frame

Defining Polarisation

We can distinguish two special cases: ( = ± )/4 - circular

( = ± )/2 - linear

When we relate the two reference frames:

These are equivalent to our initial relations:

If:

The angle ( is not really an angle; it defines the axis ratio of the ellipse and it’s ‘handedness’ (the direction E traces out elllipse), (,lies between -)/2 and +)/2. For ( > 0 the rotation is clockwise, and for (<0the rotation is counter-clockwise.

Stokes ParametersGiven #1, *1, #1, *1 these equations can be solved for #0, ( & '.

A convenient way of doing this is by means of STOKES PARAMETERSfor monochromatic waves, defined:

Thus, For completely elliptically polarized,monochromatic radiation:

In other words 3 out of 4 stokes parameters areindependent, not surprising given that theellipse of polarised radiation is fully defined by3 quantities: amplitude #, orienatation ' &handedness (.

Quasi-monochromatic polarisationIn real-life the amplitude and phase of E-field will vary with time, if this issufficiently slow, called quasi-monochromatic.

Definition of the Stokes parameters nowinvolves time averaging, and so

Stokes parameters are additive, so partially polarised light can always bedecomposed in a fully polarised and a fully unpolarised part:

We can now define the degree of polarisation, +

Using rotating polarising plate, for linear polarisation degree can bemeasured, this gives lower limit for other polarisation types.

EM PotentialsMaxwell’s Equations can also be written as two equations in terms of ascalar potential ,(r,t) and a vector potential A(r,t).

Together with definitions for E and B:

The general solution of these equations is:

RETARDED POTENTIALS

Plasma EffectsIsotropic plasmas: Dispersion

In an isotropic plasma (ie., no magnetic field) only waves can propagate with afrequency above the plasma frequency:

Such waves travel atthe phase velocity

Where, the index of refraction, nr is defined:

The phase velocity always exceeds the speed of light. However energy (andinformation) can only flow at the group velocity, vg = cnr , which is alwayssmaller than c.Because vg ! &-1, signals at different frequency travel at different speeds.This causes the pulse of a pulsar to arrive at different times on Earth. We findthe derivative of the arrival times tp to frequency to be:

Dispersion measure

Anisotropic plasmas: Faraday rotationIn an anisotropic plasma with a (tangled) magnetic field another frequencybecomes important, the cyclotron frequency:

In such a plasma the propagation speed of the waves depends on theirpolarization. A linearly polarized wave will change its angle by an amount:

Faraday Rotation

Through the frequency dependency of -. we can find B || or a lower limit to itif the magnetic field is as tangled as we think it is. If the field is so strongthat -. varies by close to 900 within the bandwidth of our measurement thewave is depolarised.