Radiative Processes
in Astrophysics
3. Basic Theory of
Radiation Fields
Eline Tolstoy
http://www.astro.rug.nl/~etolstoy/astroa07/
Lorentz Force Law
description of radiation in terms of electromagnetic theory
in the non-relativistic limit, the Lorentz force exerted on a particle with charge
q, 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 direction
of the velocity vector and the field.
e.g., providing the E field is negligible, charged particles are forced to spiral
around magnetic field lines and cannot cross them except by collisions. This
means 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 works
perpendicular to motion.
for non-relativistic
particles
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 of
increase of mechanical energy density.
Review of Maxwell’s Equations
elegant and concise way to state the fundamentals of electricity and magnetism
describes the
behaviour of both
electric and
magnetic fields as
well as their
interactions with
matter
Poynting vector
D=#E
B=µH
using Maxwell’s Eqns, consider the work done per unit volume on a particle
distribution (dotting Ampere’s law):
using
and Faraday’s law
Poynting (contd)
rate of change of total energy density
Poynting vector
The Poynting flux has some peculiar features, namely - it appears to say that
a charged bar magnet has an energy flux that circulates around the bar in a
toroidal sense, which is not very meaningful, however if one integrates the
equation over some finite volume then the rhs can be converted into an integral
over the surface volume element of Poynting flux and this is well defined and
free of peculiarities
Poynting (contd)
Integrating over volume element and using divergence theorem
The rate of change of total energy within volume,V, is equal to the net inward
flow of energy through the bounding surface, $.
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 can
contribute a finite amount to the rate of change of energy of the system. This
energy flowing in (or out) at large distances is called RADIATION.
Wave Equation
Maxwell’s Equations is a vacuum
(%=0, j=0):
A basic feature of these eqns is the existence of traveling wave solutions that
carry energy. Taking the curl of 3rd eqn & using 4th, :
Solving Wave Equation
The general solution of this wave equation,
has the form:
Where â1 & â2 are unit vectors; E0 & B0 are complex constants and k=kn is
the wave vector and & is the frequency.
This solution represents waves traveling in the n direction. By
superposing such solutions propagating in all directions and with all
frequencies we can construct the most general solution of the source free
Maxwell’s Eqns.
The first 2 tell us that both â1 and â2 are transverse to the direction of
propagation, k. With this can carry out cross products in second 2 eqns and
see that â1 and â2 are perpendicular to each other, ie., â1, â2 & k form a righthanded set.
Plane EM-waves
The values of E0 and B0 are related:
thus can show that E0 = B0
and & = ck
The Radiation Spectrum
From the TIME VARIATION OF THE ELECTRIC FIELD, and
analogously the magnetic field, follows the spectrum of the radiation.
The spectrum is the amount of energy per unit area per unit time per unit
frequency interval, and is most easily derived through a Fourier
transformation.
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 surface
element 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 shape
The fact that the time variation of the electrical field and its spectrum are related through a Fourier
transform makes it very convenient to derive a spectral shape from the characteristics of E(t)
em-pulse
radiation
spectra
a pulse of duration T has a spectrum
stretching over a bandwidth of ~1/T
A periodic signal with frequency &0 for
a duration, T will have a spectrum
width 1/T centred on &0
A similar periodic signal with a decay
time of T (damped oscillator) will produce
a spectrum of bandwidth 1/T centered on
&0, but without the higher and lower
frequency wiggles of previous example
Polarisation
mono-chromatic plane waves,
linearly polarised
Linearly polarised means the electric vector simply oscillates in direction
â1, which, with the propagation direction defines the plane of polarization.
By superimposing solutions corresponding to two such oscillations in
perpendicular we can construct the most general state of polarisation for a
wave of given k and &.
the vector E traces out an ellipse
Orthogonal components
Any (single frequency) E wave can be decomposed into 2 orthogonal
waves with amplitudes E1 and E2 and the same frequency (with different
phase). The resulting composite E traces out an ellipse.
Monochromatic polarisation
The 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 along
the^
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, and
also in the frame aligned with the principle axes of
the ellipse:
system rotated by angle ' with respect to lab frame
Defining Polarisation
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 (<0
the rotation is counter-clockwise.
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:
Stokes Parameters
Given #1, *1, #1, *1 these equations can be solved for #0, ( & '.
A convenient way of doing this is by means of STOKES PARAMETERS
for monochromatic waves, defined:
Thus,
For completely elliptically polarized,
monochromatic radiation:
In other words 3 out of 4 stokes parameters are
independent, not surprising given that the
ellipse of polarised radiation is fully defined by
3 quantities: amplitude #, orienatation ' &
handedness (.
Quasi-monochromatic polarisation
In real-life the amplitude and phase of E-field will vary with time, if this is
sufficiently slow, called quasi-monochromatic.
Definition of the Stokes parameters now
involves time averaging, and so
Stokes parameters are additive, so partially polarised light can always be
decomposed 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 be
measured, this gives lower limit for other polarisation types.
EM Potentials
Maxwell’s Equations can also be written as two equations in terms of a
scalar 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 Effects
Isotropic plasmas: Dispersion
In an isotropic plasma (ie., no magnetic field) only waves can propagate with a
frequency above the plasma frequency:
Such waves travel at
the phase velocity
Where, the index of refraction, nr is defined:
The phase velocity always exceeds the speed of light. However energy (and
information) can only flow at the group velocity, vg = cnr , which is always
smaller 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 find
the derivative of the arrival times tp to frequency to be:
Dispersion measure
Anisotropic plasmas: Faraday rotation
In an anisotropic plasma with a (tangled) magnetic field another frequency
becomes important, the cyclotron frequency:
In such a plasma the propagation speed of the waves depends on their
polarization. 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 it
if the magnetic field is as tangled as we think it is. If the field is so strong
that -. varies by close to 900 within the bandwidth of our measurement the
wave is depolarised.