blackbody radiation: planck’s law

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.

MaxMaxKarlKarl

ErnstErnstLudwigLudwigPlanckPlanck

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

1e1hc2)(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+142hc2: 3.7415343E-16 # 600 1.63E+12 7.83E+13 4.81E+14

# 650 2.02E+12 7.26E+13 3.96E+14At 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+142hc2: 3.7415343E-16 600 1.63E+12 7.83E+13 4.81E+14

650 2.02E+12 7.26E+13 3.96E+14At 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 LawET = 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 Wien

1864 -1864 - 19281928

Wien’sDisplacement

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 LawGives 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