Lecture 5 stars as black bodies
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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


Stellar spectral classification
Stellar Spectral Classification

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