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Lecture 5: Stars as Black-bodies. Objectives - to describe: Black-body radiation Wien’s Law Stefan-Boltzmann equation Effective temperature. Spectrum formation in stars is complex ( Stellar Atmospheres ) B-B radiation  simple idealisation of stellar spectra

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lecture 5 stars as black bodies

Lecture 5: Stars as Black-bodies

Objectives - to describe:

Black-body radiation

Wien’s Law

Stefan-Boltzmann equation

Effective temperature

  • Spectrum formation in stars is complex (Stellar Atmospheres)
  • B-B radiation  simple idealisation of stellar spectra
  • Usually see objects in reflected light. But:
  • all objects emit thermal radiation
  • e.g. everything here (including us!) emitting ≈ 1kW m-2
  • Thermal spectrum simplest for case of Black-body

PHYS1005 – 2003/4

planck and the black body spectrum
Planck and the Black-body spectrum
  • Black-body absorbs 100% of incident radiation (i.e. nothing reflected!)
    •  most efficient emitter of thermal radiation
  • Explaining Black-body spectrum was major problem in 1800s
    • Classical physics predicted rise to ∞ at short wavelengths  UV “catastrophe”
    • Solved by Max Planck in 1901 in ad hoc, but very successful manner, requiring radiation emitted in discretequanta (of hע), not continuously
    •  development of Quantum Physics
    • Derived theoretical formula for power emitted / unit area / unit wavelength interval:
  • where:
  • h = 6.6256 x 10-34 J s-1 is Planck’s constant
  • k = 1.3805 x 10-23 J K-1 is Boltzmann’s constant
  • c is the speed of light
  • T is the Black-body temperature

N.B. you don’t have to remember this!

PHYS1005 – 2003/4

black body spectra for different t
Black-body spectra (for different T):
  • Key features:
  • smooth appearance
  • steep cut-off at short λ(“Wien tail”)
  • slow decline at long λ(“Rayleigh-Jeans tail”)
  • increase at allλ with T
  • peak intensity: as T↑, λpeak↓(“Wien’s Law”)
  • all follow from Planck function!
  • Wien’s (Displacement) Law:
    • (math. ex.) evaluate dBλ/dλ = 0  peak λ
    • λmaxT = 0.0029
    • where λ in m and T in Kelvins.

PHYS1005 – 2003/4

example in nature of b b radiation

e.g. application of Wien’s Law:

Example in Nature of B-B radiation:
  • Space-mission called Darwin proposed to look for planets capable of harbouring life. At about what λ would they be expected to radiate most of their energy?
  • Answer:
  • N.B. we are all radiating at this λ
  • Most “perfect” B-B known!
  • What is it?
  • N.B. λ direction

PHYS1005 – 2003/4

spectra of real stars
Spectra of real stars:

T

30,000K

5,500K

3,000K

Can you cite a well-known example of any of these?

PHYS1005 – 2003/4

comparison of sun s spectrum with spica and antares
Comparison of Sun’s spectrum with Spica and Antares:

N.B. visible region of spectrum

PHYS1005 – 2003/4

spectral sequence for normal stars
Spectral Sequence for Normal Stars:

Classification runs from hottest (O) through to coolest (M)

PHYS1005 – 2003/4

power emitted by a black body
Power emitted by a Black-body:

= σ T4 / unit area

  • simply integrate over all λ
  • where σ = 5.67 x 10-8 W m-2 K-4 = Stefan-Boltzmann constant
  • e.g. what is power radiated by Sun if it is a B-B of T = 6000K?
  • Hence total L from spherical B-B of radius R is
  • Very important! Remember this equation!

L = 4 π R2σ T4

PHYS1005 – 2003/4

effective temperature t eff
Effective Temperature, Teff :
  • real stars do not have single T define Teff as
  • T of B-B having same L and R as the star

i.e. L = 4 π R2σ (Teff)4

  • e.g. Sun has L = 3.8 x 1026 W and R = 6.96 x 108 m. What is its Teff?
    • Answer: inverting above equation:
    • and inserting numbers  Teff = 5800 K (verify!)

Teff = (L / 4 π R2σ)1/4 K

Additional reading : Kaufmann (Chap. 17, 18), Zeilik (Chap. 8, 13)

PHYS1005 – 2003/4

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