-­‐  Covered thus far… -­‐  Specific Intensity, mean intensity, flux density, momentum flux -­‐  Emission and absorp>on coefficients, op>cal depth -­‐  Radia>ve transfer equa>on -­‐  Planck func>on, Planck spectrum, brightness temperature -­‐  Kirchhoff’s Law, principle of detailed balance, Einstein coefficients -­‐  ScaKering Sun from Mars Sun from Earth -­‐ Average distance of Mars from the Sun is ~1.5 AU How much longer should you expose a photo of the Sun from Mars than a photo of the Sun from Earth? Sun from Mars Sun from Earth -­‐ Average distance of Mars from the Sun is ~1.5 AU How much longer should you expose a photo of the Sun from Mars than a photo of the Sun from Earth? -­‐ Answer: The same amount of 5me! Energy flowing through unit area into unit
solid angle in unit frequency in unit time Brightness or specific intensity
Mean intensity: Flux density: Charged par>cle accelera>on Atomic and molecular transi>ons Bound-­‐bound transi>ons Bound-­‐free transi>ons Interac>on with the Coulomb fields of other charged par>cles Bremsstrahlung (free-­‐free emission) Interac>on with magne>c fields Cyclotron emission, Synchrotron emission, some coherent emissions Vibra>onal transi>ons Rota>onal transi>ons Interac>on with the E-­‐field of EM waves Thomson scaAering, Compton scaAering, Inverse Compton ScaAering -­‐  Determine emission and absorp>on coefficients and solve transfer equa>ons At radio frequencies, the emission is due
to relativistic electrons spiralling in
magnetic field lines – synchrotron
radiation
Electrons have a power law energy
distribution…
with
typically α~-0.75
High frequencies (optically thin)
Low frequencies (optically thick)
In the case of complex spectra above, multiple components of varying angular size contribute
Radio emission is produced due to
acceleration of electrons by
positive ions, i.e., free-free
emission (Bremsstrahlung) with
jνff and ανff to be derived later in
the course. For now…
Spectral index, α, defines the
frequency dependence of the
emission, such that
Spectrum rises as ν2 at low
frequencies and flattens at
higher frequencies. Why?
Recall
High frequencies (optically thin)
Low frequencies (optically thick)
D = εE
B = μH
B = magnetic flux density
E = electric field
D = electric displacement field
H = magnetic field strength
v = velocity
c = speed of light
μ = magnetic permeability
ε = dielectric constant
j = current density
ρ = charge density
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:
- the force due to the magnetic field is always perpendicular to both the
direction of the velocity vector and the field
- The rate of work done by fields on a particle is
- vŸ(vxB) = 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:
where ρ and j are the charge and current density - Rate of work done by the field per unit volume
= rate of change of mechanical energy per unit volume due to the fields
field
- Poynting Theorem
Deriviation:
(R&L 2.1)
- The rate of change of mechanical energy per unit volume plus the rate of change of
field energy per unit volume equals minus the divergence of the field energy flux
- electromagnetic field energy per unit volume
Poynting Vector
Integrating over volume element
The rate of change of total energy within
volume, V, is equal to the net inward
flow of energy
- 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.
- However… 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 electromagnetic radiation.