ASTR 498N                   Lecture 2             Déjà Review

 

(online at www.astro.umd.edu/~drabin/)

 

 

Radiative Transfer

(O&C §9.3)

 

Linear absorption coefficient (opacity)  k_ν          

Note that   [k_ν^-1] = m   and is a measure of the mean free path for photon absorption.

 

The opacity k_ν includes any process that removes energy from the frequency-specific pencil of radiation described by I_ν, whether by change in direction (scattering), change in frequency, or conversion to another form of energy.

 

Optical depth  τ_ν                                                            

 

Mass absorption coefficient  κ_ν                                

 

Cross section  σ_ν                                                             

(O&C §10.5)

 

Emission coefficient                              

 

                                     

            P    Does   ?  

 

 

Equation of radiative transfer                            

or

                                                                           

where the source function S_ν is defined as

                                                                           

 The radiative transfer equation is fundamentally probabilistic.

 

Formal solution of the radiative transfer equation:

 

                                                  

 

The solution is formal because, in general, we don’t know S_νit is part of the solution.  However, we can see why “source function” is a sensible name; and, in some cases, we can make a reasonable assumption about the form of S_ν.

 

P   O&C Problem 9.22

 

 

Black body radiation

(O&C §3.4)

 

Intensity                   (Planck function)

 

 

Wavelength of maximum B_λ        Tλ_max = 2.898e-3 m K    (Wien displacement law)

 

Frequency of maximum B_ν            Tc/ν_max = 3.670e-3 m K

 

 

 

Wien approximation (  )                        

 

Rayleigh-Jeans approximation (  )          (Note:  still B_ν)

 

 

Flux density             (Stefan-Boltzmann law and constant)

 

Stellar effective temperature T_e            

 

Source function for a gas in TE                P    Why?

 

P    Why can a star never be strictly in thermodynamic equilibrium?

 

 

Maxwell speed distribution

(O&C §8.1)

 

The density function of particle speed in a nondegenerate, nonrelativistic gas in TE is

 

                                           

 

where   gives the probability of finding a particle of mass m in the velocity range v to v + dv at temperature T.  Since it’s a probability, .

 

P    What is v_mp, the most probable speed for a particle of mass m and temperature T?      Write this also as a convenient numerical formula for electrons and for hydrogen atoms.

 

If we normalize all speeds to the most probable speed, , the density of x has a universal shape:

 

The Maxwell speed density is derived from the more fundamental 6-dimensional phase space density describing the equilibrium of an ideal gas,

 

            (Maxwell-Boltzmann)

 

where .   does not depend on the spatial coordinates x because there is no spatial (or time) dependence in a true equilibrium.  Note that this density is a pure Gaussian.

 

P   Derive the density function for the radial (line-of-sight) velocity (which governs spectroscopic Doppler shifts) and for the kinetic energy .

 

The Maxwell-Boltzmann density function is approximately applicable in a wide variety of circumstances.  However, we will find that the underlying Maxwell-Boltzmann statistics are not adequate to describe the state of all stellar interiors.   We will also need Fermi-Dirac statistics and Einstein-Bose statistics (the Planck radiation function, above, can be derived from Einstein-Bose statistics, which apply to a photon gas).

 

 

Boltzmann formula (excitation)

(O&C §8.1)

 

                                                            

 

where u (upper) and l (lower) label two energy levels of an atom or ion, n (m^-3) is the population of the level, g is the statistical weight of the level (the number of states with different quantum numbers but the same energy), and  is the energy difference between the levels.

 

The Boltzmann formula holds for thermodynamic equilibrium at temperature T.  Sometimes an excitation temperature T_ex is defined through this relationship even when TE is not known to hold, but such a temperature applies only to the specific levels involved.

 

 

Saha equation

(O&C §8.1)

 

                                            (Saha)

 

describing the relative number densities of two ionization states of the same ion, r-times ionized and (r +1)-times ionized, in terms of the ionization potential χ_r, the temperature T, and the partition functions u_{r +1} and u_r, where

 

                                             

 

The partition function is a summation of the statistical weights of all the bound states of the ion, each weighted by a Boltzmann factor to account for the relative population the level relative to the ground state.  Some authors (including O&C) use Z instead of u to denote a partition function.

 

The Saha equation is sometimes multiplied through by kT and written in terms of P_e, the electron pressure.  This assumes the perfect gas law, , and is not correct if, for example, the gas is partially degenerate.

 

 

For numerical work, it’s convenient to have the Saha equation in numerical form, e.g.,

 

                        

 

where n is in m^-3 and χ_r is in eV.  This is the origin of the odd factor 5040/T  that crops up in astronomical papers.  Notation:  “log” means log₁₀, “ln” means log_e.

 

It’s a useful fact, illustrated for atomic nitrogen below (the label above the red points should be N II, not N), that one rarely needs to consider more than two stages of ionization at once in a stellar interior or atmosphere.

 

 

 

The Saha equation applies to thermodynamic equilibrium.  As in the case of the Boltzmann formula, an ionization temperature T_i can be defined with reference to a specific ion and levels.

 

P   Derive an expression for , the population of an individual level k of an r-times ionized species relative to the total population of that ion.

 

P   O&C Problems 8.10(a), 8.10(b), and 8.(12).