Download A540 Review - Chapters 1, 5-10

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
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!)
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-theenvelope 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.
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 1.202 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.
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
10-24 gm)
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
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
dE
I 
cos dAdwdtdv
1
J 
Id

4
J is the mean intensity averaged over 4
steradians
Flux
• Flux is the rate at which energy at
frequency  flows through (or from) a unit
surface area either into a given hemisphere
or in all directions.
• Units are ergs cm-2 s-1
F   I cos d
F  2
 /2
 I sin  cosd
0
• Luminosity is the total energy radiated
from the star, integrated over a full
sphere.
• F=sTeff4
and
L=4R2sTeff4
Black Bodies
• Planck’s Law
Il 
2hc 2
l
5
1
e
hc / lkT
1
• 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)
I = 2hc2l-5 e
(-hc/lkT)
Using Planck’s Law
Computational form:
Bl (T ) 
1.19 x10 27 l5
1.44 x108 / lT
e
1
For cgs units with wavelength in Angstroms
The Solar Numbers
• F = L/4R2 = 6.3 x 1010 ergs s-1 cm-2
• I = F/ = 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!)
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
dI    I dx
– Intensity
d   rdx
– Pathlength
 r
dI   I d
• Optical depth is the integral of the absorption
coefficient times the density along the path
L
    rdx
0
I (  )  I (0)e

Radiative Equilibrium
• To satisfy conservation of energy,
the total flux must be constant at all
depths of the photosphere

F ( x)  F0   F d
0
• Two other radiative equibrium
equations are obtained by integrating
the transfer equation over solid angle
and over frequency
Convection
• If the temperature gradient
d log P


d log T   1
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.
The Transfer Equation
• For radiation passing through gas, the
change in intensity I is equal to:
dI = intensity emitted – intensity absorbed
dI = jrdx – rI dx
dI /d = -I + j/ = -I + S
• This is the basic radiation transfer equation
which must be solved to compute the
spectrum emerging from or passing through
a gas.
Solving the Gray Atmosphere
• Integrating the transfer equation over
frequency:
dI
cos 
 I S
d
• The radiative equilibrium equations give us:
F=F0, J=S, and dK/d = F0/4
• LTE says S = B (the Planck function)
• Eddington Approximation (I independent of
direction)
3
2 14
T ( )  ( (  )) Teff
4
3
Monochromatic Absorption Coefficient
• Recall d = rdx. We need to calculate , the
absorption coefficient per gram of material
• First calculate the atomic absorption coefficient a
(per absorbing atom or ion)
• Multiply by number of absorbing atoms or ions per
gram of stellar material (this depends on
temperature and pressure)
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
Hydrogen Bound-Free Absorption Coefficient
a (cm-2 per atom) x 10^6
3.5E-14
3E-14
2.5E-14
2E-14
Balmer
Absorption
1.5E-14
1E-14
5E-15
Lyman
Absorption n=1
Paschen
Absorption
n=3
n=2
0
100
600
1200 2200 3200 4200 5200 6200 7200 8200 9200
Wavelength (A)
Neutral hydrogen (bf and ff) is the dominant
Source of opacity in stars of B, A, and F
spectral type
Opacity from the H- Ion
• Only one known bound state for bound-free
absorption
• 0.754 eV binding energy
• So l < hc/h = 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)
Dominant Opacity vs. Spectra Type
Low
Electron scattering
(H and He are too
highly ionized)
He+ He
Low pressure –
less H-
Neutral H H-
H-
High
(high pressure forces more H-)
O
B
A
F
G
K
M
The T() Relation
• In the Sun, we can get the T() relation from
– Limb darkening or
– The variation of I with wavelength
– Use a gray atmosphere and the Eddington
approximation
• In other stars, use a scaled solar model:
Teff Star
T ( ) 
T ( ) Sun
Teff Sun
– 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
Hydrostatic Equilibrium
• Since d= rdx
• dP/dx= r dP/d=gr or
dP/d  = g/
The Paschen Continuum vs. Temperature
1.00E-02
50,000 K
Flux Distributions
Log Flux
1.00E-03
1.00E-04
1.00E-05
4000 K
1.00E-06
1.00E-07
300
400
500
600
700
Wavelength (nm)
800
900
1000
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 10-12 W m-2 A-1
• We can convert V magnitude to Fl:
Log Fl = -0.400V – 8.451 (erg s-1 cm-2 A-1)
Log F = -0.400V – 19.438 (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)
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:
Fbol
BC  2.5
 constant
FV
BC = mV – mbol = Mv - Mbol