Lecture 5 PHYS1005 – 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 5 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 discrete quanta (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!

Lecture 5 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 all λ 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.

Lecture 5 PHYS1005 – 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 • λmax = 0.0029 / 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 5 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?

Lecture 5 PHYS1005 – 2003/4

Comparison of Sun’s spectrum with Spica and Antares:

N.B. visible region of spectrum

Lecture 5 PHYS1005 – 2003/4

Spectral Sequence for Normal Stars:

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

Lecture 5 PHYS1005 – 2003/4

Stellar Spectral Classification

Spectral Type

Colour Teff

(K)

Spectral

Characteristics

e.g.

O UV >25,000 HeII (emission and absorption)

10 Lac

B blue 11,000-

25,000

HeI absorption, HI Rigel,

Spica

A blue-white 7,500-

11,000

HI max at A0, decreasing after

Sirius,

Vega

F white 6,000-

7,500

Metals noticeable Canopus,

Procyon

G yellow 5,000-

6,000

Solar-type, metals stronger e.g. CaI, II

Sun,

Capella

K orange 3,500-

5,000

Metals dominate Arcturus,

Aldebaran

M red <3,500 Molecular bands e.g. TiO

Betelgeuse, Antares

Lecture 5 PHYS1005 – 2003/4

Power emitted by a Black-body:

• 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?

– Answer: ~70 MW m-2

• Hence total L from spherical B-B of radius R is

• Very important! Remember this equation!

= σ T4 / unit area

L = 4 π R2 σ T4

Lecture 5 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 R2 σ (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 R2 σ)1/4 K