Basic Properties of Stars - 4 §3.4-3.6

Basic Properties of Stars - 4 §3.4-3.6Basic Properties of Stars - 4 §3.4-3.6Basic Properties of Stars - 4 §3.4-3.6Basic Properties of Stars - 4 §3.4-3.6

Colours of stars and blackbody radiationColours of stars and blackbody radiation

Colours of starsColours of starsColours of starsColours of stars

Stars have colours. Why?Stars have colours. Why?

Colours of starsColours of starsColours of starsColours of stars

Stars have colours. Why?Stars have colours. Why?Its not due to their redshift!!Its not due to their redshift!!

Blackbody RadiationBlackbody RadiationBlackbody RadiationBlackbody Radiation

OrionOrion

Surface Temp = 3600KSurface Temp = 3600KSurface Temp = 13,000KSurface Temp = 13,000KRigel: blue-Rigel: blue-white.white.

Betelgeuse: redBetelgeuse: red

Blackbody Radiation HistoryBlackbody Radiation HistoryBlackbody Radiation HistoryBlackbody Radiation History

In 1792, Thomas Wedgewood first observed that In 1792, Thomas Wedgewood first observed that all his ovens glowed “red-hot” at the same all his ovens glowed “red-hot” at the same T, regardless of size, shape or materials.T, regardless of size, shape or materials.

All objects T > 0 K emit radiation. All objects T > 0 K emit radiation. Below 800 K in the IR, 800 to 1000 K Below 800 K in the IR, 800 to 1000 K detected in optical detected in optical

By 3000 K white hot - as T goes upBy 3000 K white hot - as T goes upSpectrum shifts to shorter wavelengths & Spectrum shifts to shorter wavelengths & power increasespower increases

Perfect emitter absorbs all light it receives Perfect emitter absorbs all light it receives and reradiates it - called a blackbody.and reradiates it - called a blackbody.

Blackbody Radiation HistoryBlackbody Radiation HistoryBlackbody Radiation HistoryBlackbody Radiation History

Perfect blackbody is an Perfect blackbody is an idealization, but closelyidealization, but closely

approximated as below.approximated as below.

Cavity constant Temp Cavity constant Temp

Small hole

J. Stefan 1879 foundrelation between totalpower emitted and TP=T4

=5.67 x 10-8 W m-2 K-4

Blackbody Radiation Blackbody Radiation Blackbody Radiation Blackbody Radiation

Blackbody spectrum depends only on Blackbody spectrum depends only on

T of source.T of source.

Blackbody Radiation Blackbody Radiation Blackbody Radiation Blackbody Radiation

Blackbody spectrum depends only on Blackbody spectrum depends only on

T of source.T of source.

As T increases, As T increases, maxmax decreases and decreases and

B(B()d)d increases increases

maxmaxT = d (Wien’s Law) and d = 2.898 T = d (Wien’s Law) and d = 2.898

x 10x 10-3-3 m K. m K.

Blackbody Radiation and Quanta Blackbody Radiation and Quanta

Blackbody Radiation and Quanta Blackbody Radiation and Quanta

By 1900, Max Planck found empirical By 1900, Max Planck found empirical formula for blackbody curve.formula for blackbody curve.

BB(T) = (a(T) = (a-5-5) / e ) / e b/b/T T -1 and he tried to -1 and he tried to

derive the constants a and b.derive the constants a and b.

Problem is shown on following slidesProblem is shown on following slides

Standing wavesStanding wavesStanding wavesStanding waves

Blackbody radiationBlackbody radiationBlackbody radiationBlackbody radiation

To circumvent this problem, Planck assumed that a To circumvent this problem, Planck assumed that a standing E-M wave could not acquire any standing E-M wave could not acquire any arbitrary amount of energy, but only allowed arbitrary amount of energy, but only allowed values that were multiples of a minimum wave values that were multiples of a minimum wave energy.energy.

This quantum is given by This quantum is given by hh (or (or hhcc//)), where , where h h is is constant = 6.63 x 10constant = 6.63 x 10-34-34 J sec (Planck’s constant). J sec (Planck’s constant).

Higher Higher (shorter (shorter ) of wave, greater minimum ) of wave, greater minimum energy.energy.

Short Short , high , high waves cannot contain even 1 waves cannot contain even 1 quantum! quantum!

The Planck FunctionThe Planck FunctionThe Planck FunctionThe Planck Function BB(T) = energy emitted per second, per unit (T) = energy emitted per second, per unit wavelength interval dwavelength interval d at wavelength at wavelength , per unit , per unit area into a unit solid angle by a blackbody of area into a unit solid angle by a blackbody of temperature T (whew!)temperature T (whew!)

BB(T) = (2hc(T) = (2hc22//55)(1/e)(1/ehc/hc/ktkt - 1) w m - 1) w m-2 -2 mm-1-1 sterad sterad-1-1

Where c = speed light = 3 x 10Where c = speed light = 3 x 1088 m s m s-1-1

k = Boltzmann constant = 1.38 x 10k = Boltzmann constant = 1.38 x 10-23-23 J K J K-1-1

h = Planck’s constant = 6.63 x 10h = Planck’s constant = 6.63 x 10-34-34 J s J s

Only variable in the Planck function is T Only variable in the Planck function is T

In terms of frequency BIn terms of frequency B(T) = (2h(T) = (2h33/c/c22)(1/e)(1/ehh/kt/kt - 1) - 1) (require B(require Bdd =-B =-Bdd - as - as decreases with increasing decreases with increasing

,,dd/d/d = -c/ = -c/22, B, B =-B =-Bdd/d/d =B =Bc/c/22))

The Planck FunctionThe Planck FunctionThe Planck FunctionThe Planck Function

How does Planck function behave in the limits How does Planck function behave in the limits of very high and very low frequency? i.e hof very high and very low frequency? i.e h/kt /kt >> 1 and << 1>> 1 and << 1

Set hSet h/kT = x /kT = x B B(T) = (2h(T) = (2h33/c/c22)/(e)/(exx - 1) - 1)

For x >> 1For x >> 1, e, exx -1 = e -1 = ex x so Bso B(T) = (2h(T) = (2h33/c/c22) e) e-h-h/kT /kT

This called This called WienWien approximation approximation

For x << 1For x << 1, e, exx = 1 + x + x = 1 + x + x22/2 + x/2 + x33/6 + ……… x/6 + ……… xnn/n! /n! = 1 + x= 1 + x

Thus BThus B(T) = (2h(T) = (2h33/c/c22)(1/x) = 2kT)(1/x) = 2kT22/c/c22

Thus log BThus log B(T) = 2 log (T) = 2 log + log T + constant + log T + constant

Blackbody radiationBlackbody radiationBlackbody radiationBlackbody radiation

BB radiation - total intensityBB radiation - total intensityBB radiation - total intensityBB radiation - total intensityBB(T) = (2h(T) = (2h33/c/c22)(1/e)(1/ehh/kt/kt - 1) - 1)

The total intensity emitted by the BB is the The total intensity emitted by the BB is the integral B(T) = integral B(T) = BB(T)d(T)d = = BB(T)d(T)d = = (2h(2h33/c/c22))(1/e(1/ehh/kt/kt - 1)d - 1)d where the integral goes from 0 to where the integral goes from 0 to . .

Substitute x = hSubstitute x = h/kT, so that d/kT, so that d = (kT/h)dx. = (kT/h)dx.

Then B(T) = (2hkThen B(T) = (2hk44TT44/c/c22hh44))(x(x33/e/exx -1)dx -1)dx

Integral just a real number so that B(T) = ATIntegral just a real number so that B(T) = AT4 4 withwith A = 2kA = 2k4444/15c/15c22hh3 3

So B(T) So B(T) F F T T4 4 (F = (F = TT44). This is the ). This is the Stefan-Stefan-BoltzmannBoltzmann Law Law

= 5.67 x 10= 5.67 x 10-8 -8 w mw m-2-2 K K-4-4


Relation between Relation between maxmax and T is known and T is known

as Wien’s Lawas Wien’s Law maxmaxT= 0.002897755 m K = 0.290 cm T= 0.002897755 m K = 0.290 cm

K.K.

For a spherical For a spherical source:source:F = L / 4F = L / 4RR22, (R , (R radius circle radius circle surrounding source) surrounding source) from S-B Lawfrom S-B LawF = F = TT4 4 so L = so L = 44RR22TTee

4 4

TTe e isis the effective T the effective T

as stars are not as stars are not perfect BB radiators perfect BB radiators --T of BB that puts T of BB that puts out sameout sameenergy as the starenergy as the star


A simple problem:A simple problem:

LLsun sun = 3.839 x 10= 3.839 x 102626 W and its radius W and its radius is Ris Rsunsun = 6.955 x 10 = 6.955 x 1088 m. m.

(a) What is T(a) What is Tee of Sun? of Sun?

(b) Where does Sun’s flux peak?(b) Where does Sun’s flux peak?

(c) Any relation to the (c) Any relation to the sensitivity of human eye?sensitivity of human eye?

Basic Properties of Stars - 4 §3.4-3.6

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