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POST –MID TERM COURSE

RADIATION:

2.2.2 -Kirchof’s law-Stefan –Boltzman Law-Wein displacement law-Planck’s law-Beer’s law and radiative equilibrium-Elementary ideas of absorption, emission and scattering of radiation in the atmosphere-Solar radiation, direct and Diffuse and their measurements-Solar constant-Albedo

2.2.3 -Terrestrial radiation-Green house effect-Simpson’s diagram-Heat balance of the earth and atmosphere-Minimum and Maximum temperature

Electromagnetic Spectrum

Type of radiation Range of wavelength ( ) Range of frequency (sec-1 )

Cosmic rats, gamma rays, x-rays etc

Up to 10-3 3 x 1017 and up

Ultraviolet 10-3 to 4 x 10-1 1015 to 3 x 1017

Visible 4 x 10-1 to 8 x 10-1 4 x 1014 to 1015

Near-infrared 8 x 10-1 to 4 8 x 1013 to 4 x 1014

Infrared 4 x 102 3 x 1012 to 8 x 1013

Microwave 102 to 107 3 x 107 to 3 x 1012

Radio 107 and up Up to 3 x 107

Electromagnetic (EM) radiation is a form of energy propagated through free space orthrough a material medium in the form of electromagnetic waves.EM radiation is so-named because it has electric and magnetic fields that simultaneouslyoscillate in planes mutually perpendicular to each other and to the direction ofpropagation through space.�� Electromagnetic radiation has the dual nature:its exhibits wave properties and particulate (photon) properties.�� Wave nature of radiation: Radiation can be thought of as a travelingtransverse wave.

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Fig: A schematic view of an electromagnetic wave propagating along the zr

axis.

The electric Eur

and magnetic Huur

fields oscillate in the x-y plane and perpendicular to thedirection of propagation.

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In radiation, energy transmitted from one body to other through electromagnetic

waves without the presence of any medium and exists in discrete units or quanta in order

to explain certain observations and related by the expression shown below:

E h

c

(1)

where, c is the velocity of light (3 x 108 m/sec)

Energy of the molecule consists of Electronic, Vibrational and Rotational energy.

If the radiation falls on matter quanta of radiation and the energy is just sufficient to

elevate the electron from lower to higher orbit, then the electron will absorb this

energy. If the quanta do not match the needed for a specific transition the electron

will not absorb this energy. So for every allowed transition there are certain selection

rules, which decide the permissibility of transitions. This selective absorption will

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produce the absorption line in he spectrum. In polyatomic case there are transitions in

vibrational and rotational bands, which are bond under certain selection rules for

allowed and forbidden renditions. In reality the energy absorbed or emitted extends

over the measurable range of frequencies rather than confined to infinitesimally small

range and all the levels of the molecules are not precisely equal. Collisions between

the molecules of a gas disturb the normal energy levels and produce finite width to

the absorption line or band. In general, greatest amount of energy (highest freq) s

associated with electronic transition and intermediate amount of energy (medium

freq) and the smallest energy (least freq) are associated with rotational transitions.

Thus electronic absorptions are ordinarily found in the x-ray, UV, and visible regions

of the spectrum, vibrational absorptions in the near –infrared and rotational

absorptions in the infrared and microwave regions. But in actual the many rotational

transitions are possible in the same vibrational state and a series of rotational

absorption lines will be superimposed upon the vibrational spectrum, and this

ensemble is called rotation –vibration band.

Planck law of Radiation: States that the radiation emitted by a black body is a function of wavelength ( ) and temperature (T ). He obtained a semi-empirical expression that carried the theoretical implication that energy is quantized.

2

15

1( , )

1c

T

cf T

e

(2)

Or

2

15

( , )1

cT

c df T d E d

e

(3)

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1c and 2c are constants.

Or Intensity of radiation emitted by unit surface area into a fixed direction (solid angle)

2

5

2 1

1

hcE d d

hce kT

(4)

Where, h is Planck constant.h = 6.625 x 10-34 J-sec k = 1.38 x 10-8 J/K c = 3.0 x 108 m/sec

Stepfan –Boltzman law: States that total flux of energy (for all wavelengths) emitted by

the black body is proportional to the fourth power of absolute temperature.

Or

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2

415

0 0 0

( , )1

cT

c df T d E d T

e

(5)

Where, is Stefan Boltzmann constant.

=5.67 x 10-8 watt/m2 K4

For other hot bodies excepts the ideal radiators, the Eq-5 can be written as

0

E d

= 4e T (6)

Where, e is the emissivity of the object

Wein’s displacement Law: Sates that the wavelength of maximum emission, max as a

function of the temperature of the emitting body.

max

2897

T (7)

max is measured in microns and temperature in 0 K.

For short wave radiation, radiation emitted by the Sun (~6000 0 K)

max

2897

6000 = 0.48

0

A

For, long wave radiation (Terrestrial radiation) or radiation emitted from the Earth (T=288 0K). The maximum wavelength of emission is given by:

0

max

289710

288A

The above equation is also responsible for measuring the color temperature of the object.

Planck’s radiation law determines the effective temperature and it is the average of the

total energy emitted by the black body from black body curve. Hence, effective

temperature is always less than the color temperature of the object.

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Wein’s Radiation Law: States that, for shorter wavelengths the radiation emitted by the

black body is inversely related to fifths powers of wavelength and has exponential

dependence and energy emitted by the black body is proportional to the fifth power of

absolute temperature. From equation (4)

2

5

2 1

1

hcE d d

hce kT

for shorter wavelength, hce kT is very large and we can neglect 1 in the denominator.

2

5

2 hc

kThcE d e d

Or

5hc

kTE d e d

: (Wein’s radiation law) (7)

Or 5E d T

Rayleigh –Jeans radiation law: He proposed that for larger wavelengths (microwave

region) the radiation emitted is directly proportional to the absolute temperature and

inversely related to the fourth power of wavelength.

From equation (4)

2

5

2 1

1

hcE d d

hce kT

for larger wavelengths,

1hc

kT hce higher orders

kT

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Neglecting higher order terms, we get

2

5

2hc kTE d d

hc

Or4E d Td : (8)

In this way, we can say that from Planck law we can derive other radiations laws (like,

Wein’s displacement, Wein’s radiation and Rayleigh-Jeans law etc.) The molecules in a

gas are far apart as compared to solid and liquid molecules. Thus liquid and solids tends

to emit and absorb in very extended continuous regions of spectrum rather than discrete

lines and bands.

Kirchhoff’s law: States that, the fractional emittance a is the ratio of the incident

radiation (from the given wavelength and direction) to the total incident on it.

( , )

aE

f T

(9)

OrBlack body is having absorvity and emissivity equal or a =1. If the emission varies with

and also emission, the body in this case is called selective emitter (Gray body). Radiative transfer: When a monochromatic radiation of intensity I passes through a thin layer, ds, of gas

(Fig below) a small increment dI will be absorbed. From fractional absorption,

dI a I (10)

I I dI

ds

Fig: Absorption in an infinitesimal layer

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Measurements shows that a is proportional to the density of the gas and distance ds:

a k ds

dIk ds

I

0 0

sI s

I

dIk ds

I

00

s

k ds

sI I e

(11)

Equation –11 is the Integral form of Bee’s law for absorption. 0I is the incident

intensity sI is the intensity after penetration. Let,

k ds d Equation (11)

0sI I e

(12)

Vertical transfer at zenith angle , then

secds dz (-ve sign because we measuring +ve upwards)

secdI k I dz

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sec

0z

k dz

sI I e

(13)

In this development we have assumed that reemission is negligible compared to the

incident radiation. This case may be taken for visible light where the atmosphere is too

cool to emit any radiation. But in the case of infrared energy we cannot ignore the

reemission. From Kirchhoff’s law the emitted intensity is:

( , )E a f T : The net loss is ---

( , )dI a I a f T

Or

( , )dI f T I a (14)

This is the equation of radiative transfer, also known as Schwarzschild’s equation (in which both emission and re-emission are significant).

Terrestrial Radiation

Radiation emitted from the earth comes under long-wave and we are assuming earth as nearly black body. The mean temperature of the earth comes around +15 0C and maximum radiation comes around 10 . The maximum incoming radiation comes around

0.5 rather far in electromagnetic spectrum. As a result we can say that substances

whose radiation are not significant in short wave may become significant in long wave terrestrial region. The dominant constituents are:

Water vapour, Carbon dioxide, and Ozone. Below about 5 important amount of energy was not absorbed by the

atmosphere, partly because the atmosphere does not absorb the amounts of terrestrial radiation available for radiation. From roughly 5.0 to 8.0 there is strong absorption

band of 2H O . Beyond 8.0 the absorption by 2H O becomes smaller up to about

13.5 except for a strong band due to 3O at 9.6 , which masks a weaker band of

2CO near 10 . The strong absorption band at 13.5 and extending to 17.0 is due

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to 2CO . At its center it will absorb the terrestrial radiation completely. Again the

atmosphere is transparent beyond 17 until the rotation band of 2H O appears at 24 and beyond, the amount of terrestrial radiation at these wavelengths are secondary in importance and 14 is normally taken as cut off point beyond which the atmosphere is considered completely opaque.

Fig: Generalized diagram showing relative atmospheric radiation transmission

at different wavelengths. Blue zones show low passage of incoming and/or outgoing

radiation and white areas show atmospheric windows, in which the radiation doesn't

interact much with air molecules and hence, isn't absorbed.

Simpson diagramSimpson dealt only with the effects of 2H O and 2CO from the measurements of the

absorption stream. Let us suppose a layer in the atmosphere containing 0.03 g/cm2 of water vapour characterized as:

(1) Completely transparent below 4.0 (2) Partly transparent from 4.0 to 5.5 (3) Completely opaque from 5.5 to 7.0 (water vapour band)(4) Partly transparent from 7.0 to 8.5 (5) Completely transparent from 8.5 to 11.0 (6) Partly transparent from 11.0 to 14.0 (7) Completely opaque above 14.0 (Carbon di-oxide and water vapour bands)

In this case the surface radiates upward very nearly as black body. Part of the energy

is absorbed completely by the 2H O and 2CO in the layer of air immediately above

the surface. The rest is partly or completely transmitted. This first layer then re-

radiates at the same wavelengths and at a rate determined by its temperature.

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Solar constant:It is the amount of radiation received at the top of the atmosphere per unit area at a

perpendicular distance (mean distance from the Sun to the Earth). Its normal value is of

the order of 1368 watt/m2 .The actual direct solar irradiance at the top of the atmosphere

fluctuates by about 6.9% during a year (from 1.412 kW/m² in early January to

1.321 kW/m² in early July) due to the Earth's varying distance from the Sun, and typically

by much less than one part per thousand from day to day.

Earth receives the total amount of radiation by its cross section = 2ER and distributed

uniformly over the surface area 24 ER . . Hence at any instant the average amount of

solar radiation will be 1/4th (342 watt/m2).

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Scattering: A beam of light passing through a medium may be depleted by scattering as

well as absorption called extinction. Scattering does not involve a net transfer of radiant

energy into heat as does in absorption but merely a change in the direction with the

scattering particles. Thus a parallel beam entering the atmosphere is partly scattered to

the side and backward, so that upon reaching the surface the direct beam is less intense.

The amount of directional nature depends on size parameter 2 r

. If this ratio is

small as for air molecules and visible light then the scattering is proportional to 4

1

. This

is known as Rayleigh- Cabannes scattering. Since the shorter wavelengths are scattered

more effectively than longer wavelengths, a beam of sunlight is somewhat depleted of

blue light compared to yellow and red, which accounts for the blue of sky and the redness

of the sun near the horizon.

Mie scattering:

lies between 0.1 and 50. The dependency no longer 4

1

but varies with and

forward scattering predominate backward. The scattering of sunlight by haze, smoke,

smog and dust comes under the category of Mie scattering. Water droplets have

diameters in the range 5 μ to 100 μ. They scatter all visible and infrared wavelengths

about equally i.e. scattering is non selective with respect to wavelength. In the visible

spectrum, under conditions of non-selective scattering, equal quantities of blue, green and

red light are scattered, identification of different wavelengths becomes difficult and hence

fog and clouds generally appear white.

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Geometrical optics:When is greater than about 50, K = 2 and the angular distribution of scattered

radiation can be described by the principles of geometric optics. The scattering of visible

radiation by cloud droplets, raindrops, and ice particles falls within this regime and

produces a number of distinctive optical phenomena such as rainbows, halos, and so

forth.

(Source: Wallace and Hobbs)

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Radiation Balance:

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Appendices:

Simple model of radiation balance and green house effect:

Let the atmosphere albedo =

Surface of the planet has temperature = 0T

Temperature of the atmosphere = aT

Assuming the atmosphere as gray body with absorptivity s for shortwave and l for

longwave radiation. Kirchhoff’s law, the absorptivity and emissivity are equal.

Radiation balance at the top of the atmosphere

0(1 )S =Incoming solar radiation after reflection4

l aT = Radiation emitted by the atmosphere and escaping to space

401 l T = Radiation emitted by the surface and escaping to space (amount

absorbed 40l T )

Radiation balance at the top of the atmosphere = 4 40 01 1 l l aS T T (A)

Radiation balance at the surface

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4l aT =In coming radiation emitted by the atmosphere

40T =Outgoing radiation emitted by the surface

01 1 s S = Incoming solar radiation {note that absorptivity equals

emissivity, so the amount absorbed by the atmosphere is (1)s S}

Radiation balance at the surface = 4 40 01 1 s l aS T T (B)

Equation (A) and (B)

4 40 01 1 l l aS T T = 4 4

0 01 1 s l aS T T

On solving,

41

2l s l s

al l

ST

40

2 1

2s

l

ST

Salient features:An increase in greenhouse gasses results in an increase of l and therefore, a

monotonic increase in the surface temperature, T0. The effect of the greenhouse gasses on the atmospheric temperature is not Monotonic. At first, increasing greenhouse gasses actually decreases atmospheric temperature, while later it increases. o Though the real atmosphere is certainly more complex than.

Direct and Diffuse radiation:(http://www.atmos.millersville.edu/~adecaria/ESCI340/esci340_ra_lesson07_solar.pdf)--Source:

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Direct – This radiation comes directly from the Sun without scattering orabsorption.Diffuse – This is solar radiation that has been scattered at least once.

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Fig: Annual global energy balance for the earth atmosphere system

Radiation budget Insolation gets absorbed and heat up the earth surface some energy radiated upward in the form of OLR and some trapped in clouds which emit both ways (upward and downward).

Energy balance if there is no atmosphere:

Energy balance - no atmosphere

Incoming SW radiation = Reflected SW radiation + Outgoing longwave

If no atmosphere2R S =Incoming sort wave radiation

2R = Reflected short wave radiation2 44 R T = Outgoing longwave radiation

2 2 2 44R S R R T

4(1 ) 4S T By taking S=1370 watt/m2

α =0.3σ = 5.68 x 10-8 watt m-2

T=255 0K