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BLACKBODYBLACKBODYRADIATION:RADIATION:
PLANCK’S LAWPLANCK’S LAW
COLOR and SPECTRAL CLASSCOLOR and SPECTRAL CLASS• The light emitted by stars consists of a mixture of
all colors, but our eyes (and brain) perceive such light as being white or tinged with pastel color.
• In fact, different stars have varying amounts of each color in their light; this causes stars to have different colors.
• Most people, however, have never noticed that stars come in a variety of colors.
• When light from the Sun (or any other star) is passed through a prism, it is separated into its component colors -- a continuous spectrum.
When a beam of white light is passed through a prism, it is broken up into a
rainbow-like spectrum.
COLOR and SPECTRAL CLASSCOLOR and SPECTRAL CLASS• If the spectra of different stars are analyzed, it is
found that the intensity of the various colors differs from star to star.
• Relatively cool stars have their peak intensity in the red or orange part of the spectrum.
• The hottest stars emit blue light most strongly.• In other words, the color (or wavelength, ) of the
maximum intensity depends upon the temperature of the star.
• The star is not necessarily the color of the max-imum intensity; in fact, there are no green stars.
MaxMaxKarlKarlErnstErnst
LudwigLudwigPlanckPlanck
1858 -1858 - 19471947
• In the late 1890’s, Wien and Rayleigh had unsuccessfully attempted to formulate an equation expressing the intensity of electromagnetic radiation as a function of wavelength and the temperature of the source.
• In 1900, Planck derived the equation empirically.
• By December of 1900, Planck had derived the equation from fundamental principles.Max PlanckMax Planck
1858 -1858 - 19471947
The intensity (I) of electromagnetic radiation at a given wavelength () is a complicated function of
the wavelength and the temperature (T).
Planck’s LawIntensity of Radiation vs. Wavelength
1e
1hc2)(I
kT/hc5
2
Planck’s LawIntensity of Radiation vs. Wavelength
Planck's Constant (h): 6.6262E-34 nm 3000 5800 10000Speed of Light (c): 2.9978E+08 # 400 2.27E+11 7.42E+13 1.03E+15
Pi (): 3.1415927E+00 # 450 4.77E+11 8.22E+13 8.64E+14Boltzmann's Constant (k): 1.38066E-23 # 500 8.18E+11 8.45E+13 7.14E+14
# 550 1.21E+12 8.27E+13 5.86E+14
2hc2: 3.7415343E-16 # 600 1.63E+12 7.83E+13 4.81E+14# 650 2.02E+12 7.26E+13 3.96E+14
At 400 nm: 2hc2 / 5 3.65384E+16 # 700 2.36E+12 6.63E+13 3.27E+14
At 400 nm, 3000 K: hc / kT 11.9894485 3000 5800 10000400 0.113441 0.927449 1.029757
exp(hc / kT) 161046.5126 450 0.238543 1.027196 0.864157500 0.40892 1.055752 0.713981
exp(hc / kT) - 1 161045.5126 550 0.6073 1.033287 0.586331600 0.813035 0.978845 0.481167
I() 2.26883E+11 650 1.007968 0.907138 0.395804# 700 1.179203 0.828339 0.326925
Planck’s LawRadiation Intensity vs. Wavelength at 3000oK
(Note Peak in Infrared)
0.00E+00
5.00E+11
1.00E+12
1.50E+12
2.00E+12
2.50E+12
3.00E+12
3.50E+12
100200
300400
500600
700800
9001000
3000
Planck’s LawRadiation Intensity vs. Wavelength at 6000oK
(Note Peak in Visible)
0
2E+13
4E+13
6E+13
8E+13
1E+14
1.2E+14
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
6000
Planck’s LawRadiation Intensity vs. Wavelength at 10000oK
(Note Peak in Ultraviolet)
0
2E+14
4E+14
6E+14
8E+14
1E+15
1.2E+15
1.4E+15
100200
300400
500600
700800
9001000
10000
Planck’s LawActual Radiation Intensity vs. Wavelength at
3000, 6000, and 10000oK
0.00E+00
2.00E+14
4.00E+14
6.00E+14
8.00E+14
1.00E+15
1.20E+15
1.40E+15
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
3000
6000
10000
Planck’s LawIntensity of Radiation vs. Wavelength;Normalized Intensity vs. Wavelength
Pi (): 3.1415927E+00 450 4.77E+11 8.22E+13 8.64E+14Boltzmann's Constant (k): 1.38066E-23 500 8.18E+11 8.45E+13 7.14E+14
550 1.21E+12 8.27E+13 5.86E+14
2hc2: 3.7415343E-16 600 1.63E+12 7.83E+13 4.81E+14650 2.02E+12 7.26E+13 3.96E+14
At 400 nm: 2hc2 / 5 3.65384E+16 700 2.36E+12 6.63E+13 3.27E+14
At 400 nm, 3000 K: hc / kT 11.9894485 3000 5800 10000400 0.113441 0.927449 1.029757
exp(hc / kT) 161046.5126 450 0.238543 1.027196 0.864157500 0.40892 1.055752 0.713981
exp(hc / kT) - 1 161045.5126 550 0.6073 1.033287 0.586331600 0.813035 0.978845 0.481167
I() 2.26883E+11 650 1.007968 0.907138 0.395804# 700 1.179203 0.828339 0.326925
Planck’s LawNormalized Intensity vs. Wavelength
at 3000, 6000, and 10000 oK
0
0.2
0.4
0.6
0.8
1
1.2
3000
6000
10000
Planck’s LawNormalized Radiation Intensity vs.
Wavelength at Various TemperaturesNORMALIZED INTENSITIES
3000 4000 5000 6000 7000 8000 9000 10000100 1.78E-14 6.82E-10 2.97E-07 1.45E-05 0.0002056 0.001377 0.00569 0.016462150 2.05E-08 1.45E-05 0.000573 0.005636 0.0255825 0.072758 0.154473 0.262322200 1.44E-05 0.001377 0.016459 0.072763 0.1866222 0.345919 0.526496 0.685218250 0.000572 0.016459 0.095838 0.262286 0.4776782 0.685077 0.854446 0.948777300 0.005623 0.072761 0.262282 0.52149 0.7562274 0.914714 1.000009 1.000093350 0.02553 0.18665 0.47775 0.756354 0.9326971 0.999882 0.996075 0.925598400 0.072602 0.345935 0.685111 0.914775 0.9997807 0.980049 0.911602 0.802616450 0.152667 0.52148 0.846308 0.990501 0.9864388 0.902875 0.797008 0.673544500 0.261709 0.685109 0.948629 0.999954 0.9253141 0.802451 0.679915 0.556494550 0.388672 0.818679 0.996081 0.964589 0.8412268 0.698591 0.572807 0.457600 0.520343 0.914758 0.999937 0.902934 0.7500772 0.601214 0.479948 0.375033650 0.645099 0.973641 0.972733 0.828503 0.6609354 0.51441 0.401652 0.308499700 0.75469 0.999929 0.925454 0.750203 0.5783871 0.439165 0.336579 0.254813750 0.84446 0.999935 0.866649 0.67345 0.5043227 0.374956 0.28286 0.211535800 0.912762 0.980096 0.802491 0.601254 0.4391209 0.320629 0.238614 0.176578850 0.960102 0.946169 0.737185 0.535067 0.3823763 0.274852 0.202153 0.148237900 0.988322 0.902917 0.673439 0.475385 0.3333228 0.236329 0.17204 0.125154950 0.999951 0.854084 0.612884 0.422145 0.2910713 0.203895 0.147091 0.106255
1000 0.997754 0.802489 0.556407 0.374981 0.2547345 0.176541 0.126338 0.090697
Planck’s LawNormalized Radiation Intensity vs.
Wavelength at Various Temperatures
0
0.2
0.4
0.6
0.8
1
1.2
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
3000
4000
5000
6000
7000
8000
10000
Stefan-Boltzmann Law
ET = T4
where ET = total energy radiated per unit area over all wavelengths,
and = 5.67051 10-12 J / cm2 s K4
0
2E+14
4E+14
6E+14
8E+14
1E+15
1.2E+15
1.4E+15
ET
WilhelmWilhelmCarlCarl
WernerWernerOttoOttoFritzFritzFranzFranzWienWien
1864 -1864 - 19281928
• In 1896, Wilhelm Wien unsuccessfully attempted to derive what is now known as Planck’s Law.
• However, he did notice a relationship between the temperature of a glowing object and the wavelength of its maximum intensity of emission.
• The result of his investigation is now known as Wien’s Displacement Law.
Wilhelm WienWilhelm Wien1864 -1864 - 19281928
Wien’s
Displacement
Law:
The peak of the emission spectrum of a glowing object is a function of its temperature. The hotter the object,
the shorter the peak wavelength.
Wien’s Displacement Law
Gives max as f(T), which allows us to calculate the temperature of a star if we know the
wavelength of its maximum emission, which is easy to measure from its spectrum.
From Planck’s Law, take dI/dset = 0.
Then, maxT = 2.8979 106 nmK.
Example: max for the Sun = 502 nm.
Therefore, T = 5770K = 5500C.
The three types of Spectra:Continuous, Emission Line, and Absorption Line
Sodium Absorption Lines:The sodium vapor “subtracts out” the yellow lines
from the continuous spectrum emitted by the source.
As an excited hydrogen atom returns to its ground state, it emits the extra energy in the form of a
photon with a certain wavelength.
Each energy transition within an
atom gives rise to a
photon of a particular
wavelength.
Solar Spectrum
(Original Drawings by Fraunhofer)
Absorption lines in a
star’s spectrum reveal the
presence of elements
and compounds.
Continuous Spectrum
Absorption Spectrum of the Sun
Bright-line Spectrum of Sodium
Bright-line Spectrum
of Hydrogen
Bright-line Spectrum of Calcium
Bright-line Spectrum of Mercury
Bright-line Spectrum
of Neon
The “Inverse Square” Law: When light from a point source travels twice as far, it covers four times the
area, and is therefore only one fourth as bright.
T H E E N DT H E E N D