Lecture 5PHYS1005 – 2003/4 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

Lecture 5PHYS1005 – 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 discrete quanta (of hע), not continuously –  development of Quantum Physics –Derived theoretical formula for power emitted / unit area / unit wavelength interval: where: h = x J s -1 is Planck’s constant k = x 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!

Lecture 5PHYS1005 – 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 all λ with T peak intensity: as T↑, λ peak ↓ (“Wien’s Law”) all follow from Planck function! Wien’s (Displacement) Law: (math. ex.) evaluate dB λ /dλ = 0  peak λ λ max T = where λ in m and T in Kelvins.

Lecture 5PHYS1005 – 2003/4 Example in Nature of B-B radiation: Most “perfect” B-B known! What is it? Answer: Cosmic Microwave Background (= CMB) radiation N.B. λ direction 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: Earth T ≈ 300 K  assume this is good T for life  λ max = / 300 ≈ 10 μ i.e. well into IR. N.B. we are all radiating at this λ e.g. application of Wien’s Law:

Lecture 5PHYS1005 – 2003/4 Spectra of real stars: T 30,000K 5,500K 3,000K Can you cite a well-known example of any of these?

Lecture 5PHYS1005 – 2003/4 Comparison of Sun’s spectrum with Spica and Antares: N.B. visible region of spectrum

Lecture 5PHYS1005 – 2003/4 Spectral Sequence for Normal Stars: Classification runs from hottest (O) through to coolest (M)

Lecture 5PHYS1005 – 2003/4 Stellar Spectral Classification Spectral Type ColourT eff (K) Spectral Characteristics e.g. O UV>25,000HeII (emission and absorption) 10 Lac B blue11, ,000 HeI absorption, HIRigel, Spica A blue-white 7, ,000 HI max at A0, decreasing after Sirius, Vega F white 6,000- 7,500 Metals noticeableCanopus, Procyon G yellow 5,000- 6,000 Solar-type, metals stronger e.g. CaI, II Sun, Capella K orange 3,500- 5,000 Metals dominateArcturus, Aldebaran M red <3,500Molecular bands e.g. TiO Betelgeuse, Antares

Lecture 5PHYS1005 – 2003/4 Power emitted by a Black-body: simply integrate over all λ  where σ = 5.67 x 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? –Answer: ~70 MW m -2 Hence total L from spherical B-B of radius R is Very important! Remember this equation! = σ T 4 / unit area L = 4 π R 2 σ T 4

Lecture 5PHYS1005 – 2003/4 Effective Temperature, T eff : real stars do not have single T  define T eff as T of B-B having same L and R as the star i.e. L = 4 π R 2 σ (T eff ) 4 e.g. Sun has L = 3.8 x W and R = 6.96 x 10 8 m. What is its T eff ? Answer: inverting above equation: and inserting numbers  T eff = 5800 K (verify!) T eff = (L / 4 π R 2 σ) 1/4 K