Blackbody Radiation

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Blackbody Radiation

• Astrophysics Lesson 9

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Homework

• None today, module exams?

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Learning Objectives

• Define the term blackbody.
• Sketch and describe how shape of black body
  curves change as temperature is increased.
• Use Wiens displacement law to estimate
  black-body temperature of sources.
• Use Stefans law to estimate area needed for
  sources to have same power output as the sun.
• Recap inverse square law, state assumptions in
  its application.

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Definition of a Blackbody

• A body that absorbs all wavelengths of
  electromagnetic radiation and can emit all
  wavelengths of electromagnetic radiation.

Pure black surfaces emit radiation strongly and
in a well-defined way this is called blackbody
radiation. It is a reasonable approximation to
assume that stars behave as black bodies.
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Black Body Radiation

• The blackbody radiation of stars produces a
  continuous spectrum.
• A graph of intensity against wavelength for black
  body radiation is known as a black body curve.
• The blackbody curve is dependent on the
  temperature.

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Black Body Curve Shape
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Note the following for black bodies

• a hot object emits radiation across a wide range
  of wavelength
• As the temperature of the object increases-
• peak of the graph moves towards the shorter
  wavelengths.
• the peak is higher
• the area under the graph is the total energy
  radiated per unit time per unit surface area.

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Wiens Displacement Law

• The peak wavelength, ?max, is the wavelength at
  which maximum energy is radiated. 
• This is inversely proportional to the
  temperature, T, in Kelvin.
• This is called Wien's  Displacement Law (as the
  peak is displaced towards shorter wavelengths)-
•  
• Note the units of mK means a metrekelvin.

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Worked Example

• What is the peak wavelength of a black body
  emitting radiation at 2000 K?  In what part of
  the electromagnetic spectrum does this lie?
• ?max 0.0029 mK 2000 K
• ? max 1.45 x 10-6 m 1450 nm
• This is in the infra-red region.

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Question

• Betelgeuse appears to be red.  If red light has a
  wavelength of about 600 nm, what would the
  surface temperature be?
• Why no green stars?

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Answer

• Betelgeuse appears to be red.  If red light has a
  wavelength of about 600 nm, what would the
  surface temperature be?
• T 0.0029 mK 600 x 10-9 m 
• T 4800 K
• Why no green stars?

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Why no green stars?

• You don't get green stars because the light from
  stars is emitted at a range of wavelengths, so
  there is mixing of colours.  So those stars with
  a ?max in the green region will actually appear
  to be white.

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Luminosity of Stars

• The luminosity of a star is the total energy
  given out per second, so it's the power. 
• From the graph the luminosity increases rapidly
  with temperature, which gives rise to Stefan's
  Law. 
• The total energy per unit time radiated by a
  black body is proportional to the fourth power of
  its absolute temperature.
•  

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Stefans Law

• In other words double the temperature and the
  power goes up sixteen times.  In symbols
• L Luminosity of the star (W)
• s Stefan's constant 5.67 x 10-8 W m-2 K-4
• A surface area (m2)
• T surface temperature (K)

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Applied to Stars

• We can treat a star as a perfect sphere (A
  4pr2) and a perfect black body.  So for any star,
  radius r, we can write
•  

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Inverse Square Law

• From Earth we can measure the intensity of the
  star-
• Where L is the luminosity of the star
• d is the distance from the star

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Question

• If the Sun has a radius of 6.96 x 108 m and a
  surface temperature of about 6000 K, what is its
  total power output? 
• What is the power per unit area? 
• What is the peak wavelength?

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Answer

• L 4 p (6.96 x 108 m)2 5.67 10-8 W m-2
  K-4 x (6000 K)4
• L 4.47 1026 W
•  
• The power per unit area 4.47 1026 W 6.09
  1018 m2 7.34 107  W/m2
• Peak wavelength ?max 0.0029 mK 6000 K 4.82
  10-7 m 482 nm