Eline Tolstoykm. 195

etolstoy@astro.rug.nl

I. Definitions, Introduction toRadiative Transfer theory

Radiative Processesin Astrophysics

Electromagnetic Spectrumrefraction, diffraction & interference indicateem-radiation behaves as waves:

photo-electric effect shows energy is given to ortaken from radiation field in discrete quanta,or photons, with energy:

For thermal energy emitted by matter in thermo-dynamic equilibrium, the characteristic photonenergy is related to the temperature of the emittingmaterial:

gamma-raysX-rays

UVoptical

IR

radio

Forming a Spectrum

Dashed line shows continuous

spectrum that would have been

observed in the absence of gas.

radiationsource

Radiative Flux

When the scale of the system greatly exceeds thewavelength of radiation - we consider that the radiation

travels in a straight line (a ray) and from this factradiative transfer theory can be developed

Macroscopic Description of Radiation

describe the energy flux associated with electro-magneticradiation.

The relationship between intensity and the energy flux,momentum flux, radiation pressure and energy density.

Solid Angle

Description of a Radiation Field

The amount of energy that therays carry into cone in time dt

Average intensity of rays is defined:

For isotropic radiation field

transport of energy viaradiation: specific

intensity

The intensity provides a fairly complete description of thetransport of energy via radiation.

Energy Flux energy carried by a set of rays

( erg s-1 cm-2 Hz-1 )

Energy flux not truly intrinsic since it depends on the orientationof the surface element, energy propagating downwards is negativeenergy flux. If I! is isotropic (not a fn. of angle) then net flux = 0.

differential flux

net fluxMultiplying a function of direction by n-thpower of cos " and integrating is often called“taking the n-th moment”.

Net flux is the first moment of the intensity

Momentum Flux Photons carry momentum

the momentum flux is the second moment of the intensity

Differential momentum flux is dF! cos " and integrating overall directions gives

, Unit vector in direction of photonmotion

component of momentum in the direction normal to the surfaceelement is

Inverse square law for energy fluxIsotropic radiation:

emits energy equally in all directions (e.g., a star)

ss1 r

r1

If we can regardsphere s1 as fixed

CONSERVATION OF ENERGY

Specific Energy Density

To determine constant of proportionality the energy enclosed:

will pass out of cylinder in time, dt = dl /c

From our definition of specific intensity:

Integrating over direction:

Energy density is proportional to the zeroth moment of intensity

is the mean intensity

Radiation Pressure

For isotropic radiation, the radiation pressure is:

transfer of momentum:

time between reflections:

Rate of transfer of momentum to the wall perunit area, or radiation pressure

for isotropic radiation:

Each photon transfers twice the normalcomponent of momentum

Integrating over 2# steradians, I! = J!

Radiation pressure of an isotropic radiation field is 1/3 the energy density

Constancy of Specific IntensityConsider the energy carried by that set of rays passing through both dA1 dA2

thus the intensity is constant along a ray

Another way of stating this is by thedifferential relation

where ds is a differentialelement along the ray

Radiative Transfer

Now we consider radiation passing through matter,which may absorb, emit and/or scatter radiation intoor out of our beam. We will derive the equationgoverning the evolution of the intensity

Radiative Transfer Equation

Intensity is conserved along a ray

Unless there is emission or absorption

With emission coefficient, j! in W m-3 sr-1 Hz-1

and absorption coefficient, "! in m-1

The Equation of Radiative Transfer

In deriving RTE it is implicitly assumed that radiation propagates likeparticles. It thus neglects the wave nature of EM radiation, and treatsneither reflection, refraction nor diffraction. There are manyastrophysical situations where this is inappropriate, particularly at radiowavelengths.

There are many ways to solve RTE depending upon assumptions.

Emission

emission coefficient

for isotropically emitting particles

is the radiated power per unit volume per unit frequency

We define an emission coefficient such that matter in a volume element, dVadds to the radiation field an amount of energy, dE.

The (angle averaged) EMISSIVITY $! is the energy per unit mass per unittime per unit frequency, and is defined, such that:

with % the mass density, from which follows the relation,for isotropic emission

In going a distance ds, a beam of cross section dAtravels through a volume dV=dAds, thus the intensityadded to the beam by spontaneous emission is:

Absorptionabsorption will remove from the beam an amount of intensity proportional tothe incident intensity and the path length, ds.

For a simple model of randomly placed absorbing spheres with a cross-section &and number density n the mean covering factor for objects in a tube of area Aand length ds is dA/A = n&dV/A = n&dl so the attenuation of intensity:

where the absorption coefficient, ', has units (length)-1

Solutions to two simple limiting cases:emission only, '! = 0

absoprtion only, j! = 0

Increase in brightness is equal to emission coeff integrated along los

brightness decreases by exponential of absorption coeff integrated along los

Optical Depth & Source FunctionThe RTE takes a particularly simple form if we replace path length, s byoptical depth, (!

In terms of pure absorption:

A medium is said to be optically thick, or opaque when (! integrated along atypical path through the medium > 1

When (! < 1 then the medium is said to be optically thin or transparent

Thus the formal solution of the RTE is:

Constant source function

as ( !! then I! ! S!

Constant source function

Mean Free Path

Consider a single atom of radius 2a0 moving with speed v through a collectionof stationary points that represent the centres of other atoms:

collision cross-section of the atom

within volume, V, are nV = n&vt point-likeatoms with which the moving atom has collided,thus the average distance between collisions:

The mean free path is the reciprocal of the absorption coeff, '! for ahomogeneous material.

Radiation ForceIf a medium absorbs radiation then the radiation exerts a force on the medium,because radiation carries momentum.

radiation flux vector

A photon has momentum E/c, so the vector momentum per unit area per unittime per unit path length absorbed by medium is:

Scattering

When scattering is present, solution of RTE becomes more difficultbecause emission into d) depends on I! in solid angles d)’integrated over d) (ie, scattering from d)’ into d)).

RTE becomes an integro-differential equation which mustgenerally be solved by numerical techniques

(Isotropic) ScatteringEmission coefficient for coherent, isotropic scattering can be found simply byequating the power absorbed per unit volume and frequency ranges tocorresponding power emitted:

where &! is the SCATTERING COEFFICIENT

Dividing by &! we find the source function for scattering equals meanintensity of the emitting material:

Transfer equation for pure scattering:

Cannot use formal soln to RTE since source function is not known a prioriand depends on solution to In at all directions through a given point. It isnow an INTEGRO-DIFFERENTIAL EQUATION, which is difficult to solve,need to find approximate method of treating scattering problems.

Random walks Net displacement of photon after N freepaths:

Mean vector displacement vanishes

Now all the cross terms like vanish since the directions ofdifferent path segments are presumed uncorrelated.

but mean square displacement traveledby photon:

As an example, consider the escape of a photon from a cloud of size L. If themean free path is then the optical depth of the cloud is ( ~ L/l. In Nsteps the photon will travel a distance . Equating this with thesize of the cloud L yields the required number of steps for escape, N ~ (2 foroptically thick region, for optically thin, 1 - e*( + (, so N~(

Radiative Diffusion: Rosseland ApproxDerive a simple expression for the energy flux, relating it to the localtemperature gradient - called Rosseland Approximation.

First assume that the material (temperature,absorption coeff etc.) depend on depth in themedium - called Plane-Parallel assumption.

Convenient to use µ = cos "

Therefore, transfer Eqn: