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  Radiative Transfer, Physics tutorial

Intensity and Flux:

Intensity and flux are two essential measures of energy flow. Intensity is energy flow in the particular direction through unit area per unit time per unit solid angle, where unit area is perpendicular to the given direction stated by solid angle. Flux is net energy flow through the unit area per unit time, summed over all directions, where unit area is fixed as being perpendicular to the direction in which net flow of radiation is proceeding. Therefore, flux is attained by summing intensity over all directions.

For the isotropic intensity, positive and negative contributions will cancel out reducing flux to zero. Quantitatively, flux F and the intensity I are associated by expression

F = O'IcosqdW

Here dW is element of solid angle, and q is the angle between the fixed direction stated by flux and variable direction stated by intensity.

Intensity is estimated per unit area perpendicular to line of sight. Flux is estimated per unit area perpendicular to fixed direction of net flow of energy.

Scientists are generally interested in monochromatic intensities and fluxes like intensity per unit frequency interval at frequency v. Such specific intensities and fluxes are signified by In and Fn respectively. Quantities I and In are associated by given integral:

I = O'Indn

By solving for intensity as function of direction and integrating, you get net flow of radiation (i.e. flux) proceeding from stellar atmosphere or gas cloud.

Luminosity L:

Luminosity L of the star is closely associated to flux F. Consider the spherical star of radius R. If flux estimated at surface is F, then luminosity L is provided by:

L = 4pR2F

Therefore, for point object at a distance d, luminosity is provided by L = 4pd2F

So that flux becomes F = L/4pd2

In case of the extended object, resolved in angle by telescope, intensity received from different parts of the body is known as surface brightness of object. If object has uniform surface brightness, then intensity is independent of distance.

Radiative Transfer Equation:

Consider the pencil of radiation of frequency n travelling in atmosphere. As radiation travels a distance ds, its specific intensity changes according to expression.

dIn = ends - knds

Where εv is emission coefficient, kv is absorption coefficient and Iv is specific intensity.

The optical depth tn along the direction r is stated by equation

dtn = -kndr Here r makes angle q with ds. Now, you can write

-kn = dtn/dr

Presence of projection factor cosq signifies that

dr = cosqds

Multiplying across by cosq, we get

cosq(dIn/dtn) = en(dr/dtn) + In Now it is easy to see that dr/dtn = -1/kn

Using equation we get

cosq(dIn/dtn) = -en/kn + In or m(dIn/dt) = In - en/kn

Where projection factor

m = cosq

Equation is basic equation of radiative transfer. It has with the given sign convention:

(i) In is positive outwards (towards the observer).

(ii) tn is positive inwards (starting from zero at surface nearest the observer).

(iii) s is positive inwards (staring from zero at surface nearest the observer).

Ratio of emission coefficient to absorption coefficient states source function Sn. Therefore, source function Sn is provided by:

Sn = en/kn

Here εv and kv are emission and absorption coefficients respectively.

Local Thermodynamic Equilibrium (LTE):

In stellar atmosphere, it is not possible to get total thermodynamic equilibrium. If there were complete thermodynamic equilibrium, temperature would be similar everywhere with no temperature gradient to drive the outward flow of radiation. Further, if temperature were similar everywhere, then radiation field would be isotropic so that positive contributions and negative contributions would negate, resulting in zero flux. Undoubtedly, a total thermodynamic equilibrium can't take place in stellar atmosphere. Though, an adequately small region of stellar atmosphere may achieve roughly the same temperature. In this situation, affected region may be characterized by single local temperature. Such a small region is said to be in local thermodynamic equilibrium (LTE). LTE approximation really simplifies solution of equation of radiative transfer. For example, when LTE reigns, ratio en/kn relies only on temperature T, and source function Sn is simply Planck function Bn(T).

Planck's law can also be written in terms of wavelength l . To do this, we need that

Bndn = -Bldl or dn/dl = -Bl/Bn

Negative sign points out that wavelength decreases with increasing frequency.

Now, velocity of light is equivalent to the product of frequency and wavelength. Therefore, c = nl or

 n = cl-1

Differentiating above expression, you find that dn/dl = -cl-2

After solving we get -Bl/Bn = -cl-2

Bl(T) = (2hc2/l5)(1/[ehc/(lkBT) - 1])

Application to Radio Waves:

The main use of radio Wave is to communicate information from one place to another through intervening media (that is, air, space, and non-conducting materials) without wires. Besides being utilized for transmitting sound and television signals, radio wave is utilized for transmission of data in coded form. In form of radar it is utilized also for sending out signals and picking up their reflections from objects in their path. Long-range radio signals allow astronauts to communicate with earth from moon and carry information from space probes as they travel to distant planets. For navigation of ships and aircraft radio range, radio compass (or direction finder), and radio time signals are extensively utilized. Radio signals sent from global positioning satellites can also be utilized by special receivers for exact indication of position. Digital radio, both satellite and terrestrial gives enhanced audio clarity and volume. Different remote-control devices, comprising rocket and artificial satellite operations systems and automatic valves in pipelines, are turned on by radio signals. Development of transistor and other microelectronic devices led to growth of portable transmitters and receivers. Cellular and cordless telephones are really radio transceivers. Several telephone calls regularly are relayed by radio rather than by wires; few are sent through radio to relay satellites. Few celestial bodies and interstellar gases emit comparatively strong radio waves which are seen with radio telescopes made up of very sensitive receivers and large directional antennas.

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