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Lecture 1/2-Emission mechanisms (Specific intensity and surface brightness…
Lecture 1/2-Emission mechanisms
Specific intensity and surface brightness
Specific intensity is the power leaving unit area dA of an emitting region, at a frequency v per unit frequency interval dv, into a solid angle d omega heading towards the collector
Diagram 1
Radio photons leaing an extended emitting system
A few of these photons are heading towards the collector system and constitute an energy flow towards it
vast majority miss
astronomical sources are effectively at infinity so 3-d objects appear as 2-d surfaces
the energy stream can be described in terms of plane waves fronts or rays
Surface brightness is the power arriving per unit area da of the collector system at a frequency v per unit frequency interval dv from solid angle d Omega of the emitting region
Diagram 2
Area dA of apparent surface of source
Iv is a fundamental property related to the physics of the emission process
Surface brightness translates directly to the physical condition at the source, since Bv is independent of distance, i.e conserved (Equivalence)
Equivalence caveats
Absorption of photons by material within the beam
scattering of photons out of the beam
change of frequency: in the expanding universe frequencies fall by (1+z)
Diffraction effects: assumed geometrucal optics, antenna beam will have sidelobes pointing in different directions towards region most likely with different brightness
Solid Angles
Definition of angle in radian measure is the arc of a circle divided by its radius
Solid angle Omega is the area A on the surface of a sphere divided by its radius r squared
mathematically, solid angle does not have units but assigned steradians or sr
Black body radiation
Plank function
perfect thermal radiator or absorber
Rayleigh jeans approximation
maths
Effective brightness temperature
If using the plank formula or RJ approximation, assumed source is a perfect blackbody
most sources not black bodies, but convenient reference
Brightness temperature is the temp if source were blackbody, given its brightness Bv
If brightness temp more than 10^5K, it is not a thermal source. From non thermal particle population
Larmor's theorem
Gives the amount of energy lost by a single accelerating charged particle (usually an electron)
maths
Parseval's theorem
Fourier relationship between time and frequency
power can be function of time or frequency
Emitted energy
Energy emitted by a single electron per unit band width over 4pi steradians
Emission co-efficient
For a charged particle to be accelerated it has to experience a force
caused by something, an interaction
to calculate average energy from a population of charged particles, we need to multiply by the number of interaction that happened per unit time and per unit volume
Thh specific intensity producted per unit volume through a medium
Absorption coefficient
The specific intensity lost per unit distance through a medium
as photons pass through a plasma they will interact with the atoms or molecules in the plasma
they will be absorbed, scattered, re-emitted
some photons will not make it all the way through the plasma
Thermodynamic equilibrium
situation where the energy exchange between the radiative and kinetic energies of a gas is efficient enough that both can be described by a single temperature
can be approximated to local thermodynamic equilibrium if system is almost
In LTE if it can be characterised by a single thermal temperature
Not in LTE if the local kinetic temperature is not equal to the planckian temperature of the radiation field
Optical depth
Integrated absorption along the line of sight through the medium
dimensionless
Flux density
for sources smaller than the beam need to use flux density rather than surface brightness
flux density is the average of the surface brightness over the solid angle of the beam
since the surface brightness does not vary with distance, as the source distance increase the flux density will fall due to the smaller surface area
Sources get smaller not fainter
units of jansky