1. A540 Review - Chapters 1, 5-10 Basic physics Boltzman equation
Saha equation
Ideal gas law
Thermal velocity distributions
Definitions
Specific/mean intensities
Flux
Source Function
Optical depth
Black bodies
Planck’s Law
Wien’s Law
Rayleigh Jeans Approx.
Gray atmosphere
Eddington Approx.
Convection
Opacities
Stellar models
Flux calibration
Bolometric Corrections

2. Basic Assumptions in Stellar Atmospheres
Local Thermodynamic Equilibrium
Ionization and excitation correctly described by the Saha and Boltzman equations, and photon distribution is black body
Hydrostatic Equilibrium
No dynamically significant mass loss
The photosphere is not undergoing large scale accelerations comparable to surface gravity
No pulsations or large scale flows
Plane Parallel Atmosphere
Only one spatial coordinate (depth)
Departure from plane parallel much larger than photon mean free path
Fine structure is negligible (but see the Sun!)

3. Basic Physics – the Boltzman Equation
Nn = (gn/u(T))e-Xn/kT
Where u(T) is the partition function, gn is the statistical weight, and Xn is the excitation potential. For back-of-the-envelope calculations, this equation is written as:
Nn/N = (gn/u(T)) x 10 –QXn
Note here also the definition of
Q = 5040/T = (loge)/kT
with k in units of electron volts per degree, since X is in electron volts. Partition functions can be found in an appendix in the text.

4. Basic Physics – The Saha Equation
The Saha equation describes the ionization of atoms (see the text for the full equation). For hand calculation purposes, a shortened form of the equation can be written as follows
N1/ N0 = (1/Pe) x x 109 (u1/u0) x T5/2 x 10–QI
Pe is the electron pressure and I is the ionization potential in ev. Again, u0 and u1 are the partition functions for the ground and first excited states. Note that the amount of ionization depends inversely on the electron pressure – the more loose electrons there are, the less ionization there will be.

5. Basic Physics – Ideal Gas Law
PV=nRT or P=NkT where N=r/m
P= pressure (dynes cm-2)
V = volume (cm3)
N = number of particles per unit volume
r = density of gm cm-3
n = number of moles of gas
R = Rydberg constant (8.314 x 107 erg/mole/K)
T = temperature in Kelvin
k = Boltzman’s constant (1.38 x 10–16 erg/K)
m = mean molecular weight in AMU (1 AMU = 1.66 x gm)

6. Basic Physics – Thermal Velocity Distributions
RMS Velocity = (3kT/m)1/2
Velocities typically measured in a few km/sec
Mean kinetic energy per particle = 3/2 kT

7. Specific Intensity/Mean Intensity
Intensity is a measure of brightness – the amount of energy coming per second from a small area of surface towards a particular direction
erg hz-1 s-1 cm-2 sterad-1
Jn is the mean intensity averaged over 4p
steradians

8. Flux
Flux is the rate at which energy at frequency n flows through (or from) a unit surface area either into a given hemisphere or in all directions.
Units are ergs cm-2 s-1
Luminosity is the total energy radiated from the star, integrated over a full sphere.
F=sTeff4 and L=4pR2sTeff4

9. Black Bodies Planck’s Law
Wien’s Law – Il is maximum at l=2.9 x 107/Teff A
Rayleigh-Jeans Approx. (at long wavelength)
Il = 2kTc/ l 4
Wien Approximation – (at short wavelength)
In = 2hc2l-5 e (-hc/lkT)

10. Using Planck’s Law Computational form:
For cgs units with wavelength in Angstroms

11. The Solar Numbers F = L/4pR2 = 6.3 x 1010 ergs s-1 cm-2
I = F/p = 2 x 1010 ergs s-1 cm-2 steradian-1
J = ½I= 1 x 1010 ergs s-1 cm-2 steradian-1
(note – these are BOLOMETRIC – integrated over wavelength!)

12. Absorption Coefficient and Optical Depth
Gas absorbs photons passing through it
Photons are converted to thermal energy or
Re-radiated isotropically
Radiation lost is proportional to
Absorption coefficient (per gram)
Density
Intensity
Pathlength
Optical depth is the integral of the absorption coefficient times the density along the path

13. Radiative Equilibrium
To satisfy conservation of energy, the total flux must be constant at all depths of the photosphere
Two other radiative equibrium equations are obtained by integrating the transfer equation over solid angle and over frequency

14. Convection If the temperature gradient
then the gas is stable against convection.
For levels of the atmosphere at which ionization fractions are changing, there is also a dlogm/dlogP term in the equation which lowers the temperature gradient at which the atmosphere becomes unstable to convection. Complex molecules in the atmosphere have the same effect of making the atmosphere more likely to be convective.

15. The Transfer Equation
For radiation passing through gas, the change in intensity In is equal to:
dIn = intensity emitted – intensity absorbed
dIn = jnrdx – knrIn dx
dIn /dtn = -In + jn/kn = -In + Sn
This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

16. Solving the Gray Atmosphere
Integrating the transfer equation over frequency:
The radiative equilibrium equations give us:
F=F0, J=S, and dK/dt = F0/4p
LTE says S = B (the Planck function)
Eddington Approximation (I independent of direction)

17. Monochromatic Absorption Coefficient
Recall dtn = knrdx. We need to calculate kn, the absorption coefficient per gram of material
First calculate the atomic absorption coefficient an (per absorbing atom or ion)
Multiply by number of absorbing atoms or ions per gram of stellar material (this depends on temperature and pressure)

18. Physical Processes
Bound-Bound Transitions – absorption or emission of radiation from electrons moving between bound energy levels.
Bound-Free Transitions – the energy of the higher level electron state lies in the continuum or is unbound.
Free-Free Transitions – change the motion of an electron from one free state to another.
Scattering – deflection of a photon from its original path by a particle, without changing wavelength
Rayleigh scattering if the photon’s wavelength is greater than the particle’s resonant wavelength. (Varies as l-4)
Thomson scattering if the photon’s wavelength is much less than the particle’s resonant wavelength. (Independent of wavelength)
Electron scattering is Thomson scattering off an electron
Photodissociation may occur for molecules

19. Neutral hydrogen (bf and ff) is the dominant
Source of opacity in stars of B, A, and F
spectral type

20. Opacity from the H- Ion
Only one known bound state for bound-free absorption
0.754 eV binding energy
So l < hc/hn = 16,500A
Requires a source of free electrons (ionized metals)
Major source of opacity in the Sun’s photosphere
Not a source of opacity at higher temperatures because H- becomes too ionized (average e- energy too high)

21. Dominant Opacity vs. Spectra Type
Low
Electron scattering
(H and He are too
highly ionized)
Low pressure –
less H-
Electron Pressure
He+ He
Neutral H H- H-
High
(high pressure forces more H-)
O B A F G K M

22. The T(t) Relation In the Sun, we can get the T(t) relation from
Limb darkening or
The variation of In with wavelength
Use a gray atmosphere and the Eddington approximation
In other stars, use a scaled solar model:
Or scale from published grid models
Comparison to T(t) relations iterated through the equation of radiative equilibrium for flux constancy suggests scaled models are close

23. Hydrostatic Equilibrium
Since dtn=kn rdx
dP/dx=kn r dP/dtn=gr or
dP/dt n = g/kn

24. The Paschen Continuum vs. Temperature
50,000 K
4000 K

25. Calculating Fl from V Best estimate for Fl at V=0 at 5556A is
Fl = 3.54 x 10-9 erg s-1 cm-2 A-1
Fl = 990 photon s-1 cm-2 A-1
Fl = 3.54 x W m-2 A-1
We can convert V magnitude to Fl:
Log Fl = V – (erg s-1 cm-2 A-1)
Log Fn = V – (erg s-1 cm-2 A-1)
With color correction for 5556 > 5480 A:
Log Fl =-0.400V –8.451 – 0.018(B-V) (erg s-1 cm-2 A-1)

26. Bolometric Corrections
Can’t always measure Fbol
Compute bolometric corrections (BC) to correct measured flux (usually in the V band) to the total flux
BC is usually defined in magnitude units:
BC = mV – mbol = Mv - Mbol