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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
- 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

- 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):

- 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

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:

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:

N.B. visible region of spectrum

PHYS1005 – 2003/4

Spectral Sequence for Normal Stars:

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

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Stellar Spectral Classification

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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, 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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