Lecture 5: Stars as Black-bodies

1. 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

2. 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

3. 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

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

5. 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

6. Comparison of Sun’s spectrum with Spica and Antares: N.B. visible region of spectrum PHYS1005 – 2003/4

7. Spectral Sequence for Normal Stars: Classification runs from hottest (O) through to coolest (M) PHYS1005 – 2003/4

8. Stellar Spectral Classification PHYS1005 – 2003/4

9. 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

10. 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