Line%20Broadening%20and%20Opacity

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Line Broadening and Opacity
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Line Broadening and Opacity
Total Absorption Coefficient ?c ?l

• Absorption Processes Simplest Model
• Photon absorbed from forward beam and reemitted
  in arbitrary direction
• BUT this could be scattering!
• Absorption Processes Better Model
• The reemitted photon is part of an energy
  distribution characteristic of the local
  temperature
• B??(Tc) gt B?(Tl)
• Some of B?(Tc) has been removed and B?(Tl) is
  less than B?(Tc) and has arbitrary direction.

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Doppler Broadening
Non Relativistic ??/?0 v/c

• The emitted frequency of an atom moving at v will
  be ?' ?' ?0 ?? ?0 (v/c)?0
• The absorption coefficient will be
• Where ? Frequency of Interest
• ?' Emitted Frequency
• ?0 Rest Frequency

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Total Absorption
Per atom in the unit frequency interval at ?

• Multiply by the fraction of atoms with velocity v
  to v dv The Maxwell Boltzman distribution
  gives this
• Where M AM0 mass of the atom (A atomic
  weight and M0 1 AMU)
• Now integrate over velocity

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This is a Rather Messy Integral
Doppler Equation
Gamma is the effective ?
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Simplify!
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Our Equation is Now

• Note that ??0 is a constant (?0/c) (?2kT/M) so
  pull it out
• What about dv dv (c/?0)d(??)
• y ??/??0
• dy (1/??0) d(??)
• dv (c??0/?0)dy
• So put that in

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Getting Closer
Constants First the constant in the integral
So then
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Continuing On

• Note the a0 does not depend on frequency (except
  for ?0 constant for any line ??0
  (?0/c)?(2kT/M)). It is called the absorption
  coefficient at line center.
• The normalized integral H(a,u) is called the
  Voigt function. It carries the frequency
  dependence of the line.

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Pulling It Together
The Line Opacity al
Be Careful about the 1/vp as it can be taken up
in the normalization of H(a,u).
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The Terms

• a is the broadening term (natural, etc)
• a d'/??0 (G/4p)/((?0/c)?(2kT/M))
• u is the Doppler term
• u ?-?0/((?0/c)?(2kT/M))
• At line center u 0 and a0 is the absorption
  coefficient at line center so H(a,0) 1 in this
  treatment. Note that different treatments give
  different normalizations.

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Wavelength Forms

• G's are broadenings due to various mechanisms
• e is any additional microscopic motion which may
  be needed.

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Simple Line Profiles

• First the emergent continuum flux is
• The source function S?(t?) B?(t?).
• E2(t?) the second exponential integral.
• The total emergent continuum flux is
• t? refers to the continuous opacities and tl to
  the line opacity

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The Residual Flux R(??)

• This is the ratioed output of the star
• 0 No Light
• 1 Continuum
• We often prefer to use depths D(??) 1 - R(??)

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The Equivalent Width

• Often if the lines are not closely spaced one
  works with the equivalent width

• W? is usually expressed in mÅ
• W? 1.06(??)D(??) for a gaussian line. ?? is the
  FWHM.

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Curve of Growth
Damping/?
Log (W/)
Saturation
Linear
Log Ngf
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The Cookbook
To Compute a Line You Need

• Model Atmosphere (tR, T, Pgas, Ne, ?R)
• Atomic Data
• Wavelength/Frequency of Line
• Excitation Potential
• gf
• Species (this specifies U(T) and ionization
  potential)
• Abundance of Element (Initial)

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What Do You Do Now

• tR,?R Optical depth and opacity are specified at
  some reference wavelength or may be Rosseland
  values.
• The wavelength of interest is somewhere else ?
• So you need a? You need T, Ne, and Pg and how
  to calculate f-f, b-f, and b-b but in computing
  lines one does not add b-b to the continuous
  opacity.

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Next

• Use the Saha and Boltzmann Equations to get
  populations
• You now have t?
• Now compute tl by first computing al and using
  (a,u,??0) as defined before. Note we are using
  wavelength as our variable.