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Radiation

In this module we have been concerned with how to detect stellar objects.  To detect them we have to detect radiation emitted from them.  We are now going to consider an idealised “model” for the emission of radiation.  By “radiation” I mean any form of e/m radiation: Gamma through to Radio – including visible light.

Black body radiation

Q: what is a better emitter of radiation – matt black surfaces or polished reflectors? - matt black: any body that maintains a constant temperature will be emitting radiation at the same rate as it is absorbing it. 

Matt black is both a good absorber and a good emitter

Polished silver is a poor absorber and a poor emitter.

No surface is a 100% perfect absorber/emitter.  The IDEAL model is a “black body” – a perfect absorber and emitter.  The radiation it would emit is called BLACK BODY RADIATION.

This is a useful mathematical construction – it allows us to ignore surface properties of the radiator.  It’s use is akin to the IDEAL GAS.  Neither exist – but they allow us to do the maths.

Experimentally a black body can be improvised by taking a hollow ball – coating the walls with soot and then drilling a hole in it.  If we warm this object – the radiation it emits will simulate perfect black body radiation.

Quick practical:  1.  Observe the colour changes when running a filament lamp at different temperatures.  All wavelengths = 3000K .  Red colour = 1200K

2.  Heat nichrome wire in a Bunsen burner –  observe the colour changes.

 

 


 

Black body spectra:

See diagram P.18 – McGilivray  - note that the peaks are displaced to shorter wavelenghts with an increase in temperature.  As T is increased the intensity of each wavelength increases, but shorter wavelengths show the most marked increase.

Note: l max = approx: 3000nm  for  1000K temp  l maxT = approx = 0.003 m K

And:    l max = approx: 2000nm  for  1400K temp  l maxT = approx = 0.0028 m K  

This leads to  Wein’s displacement law:  

 Wien’s constant

l maxT  =  constant  =  0.0029 m K   Wien’s constant

This can be used to estimate the temperature of a star or other stellar objects (- “sources”) 

Estimating the Temperature of a Star

Stars approximate to black body radiators.  The wavelength that corresponds to the peak power output provides an approx method of  determining it’s effective surface temperature.

Eg:  The peak wavelength of the sun is 490 nm – estimate it’s effective surface temp:

Teff  =  0.0029 / 490 x10-9  =  5920 K – what’s that in degrees?  5920 - 273  =  5647oC

Note:

1.  assume it’s a perfect black body

2.  there is considerable variation in temp throughout the layers of a star.  The corona of the sun has a temp of 106 K while the temp at the core is 1.4 x 107 K.  Calculate the peak wavelength for these two.

Colour of stars:

Our sun is yellow cos the peak wavelength – 490 nm   - blue/ green region combines with the longer wavelengths from yellow, orange and red parts to produce the overall yellow effect.

Estimate these temps:  Betelgeuse in the constellation of Orion is a Red Giant , Sirius – the brightest star in the sky is blue in colour.

Stefan’s Law

Wein says that the total energy emitted per second increases with increasing temperature.  Increased area also increases the total energy per second emitted.  The link between radiated power and temperature was investigated by Stefan in1884.

Stefan’s law:    P  =  s A T4

P = radiated power

A = surface area

T = surface temperature

s = Stefan's constant = 5.7 x 10-8 W m-2 K-4

 

Example:

Estimate the surface temperature of the sun if has a radius of  6.96 x 108  m and it emits radiation at a power of 4.5 x 1026 Watts

--> T = 4th root of (P/ s 4 p R2) --> 6010 K 

NB not the same as Wein’s result  … assumptions based on the sun being a perfect black body.

Q  Estimate the area needed for a source at 4000K to have the same power output as the sun:

Take Tsun = 6000K

Tsun4 Asun = Tsource4Asource

Q Estimate the power radiated from the surface of the earth.  Take the average surface temperature of the earth to be 17oC

Q 9.12 & 9.13 from Breithaupt

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