A540 Review - Chapters 1, 5-10 Basic physics Boltzman equation Saha equation Ideal gas law Thermal...
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Transcript of A540 Review - Chapters 1, 5-10 Basic physics Boltzman equation Saha equation Ideal gas law Thermal...
A540 Review - Chapters 1, 5-10A540 Review - Chapters 1, 5-10
Basic physicsBoltzman equationSaha equation Ideal gas lawThermal velocity
distributions Definitions
Specific/mean intensities
FluxSource FunctionOptical depth
Black bodiesPlanck’s LawWien’s LawRayleigh Jeans
Approx. Gray atmosphere
Eddington Approx. Convection Opacities Stellar models Flux calibration Bolometric
Corrections
Basic Assumptions in Stellar Basic Assumptions in Stellar AtmospheresAtmospheres
• 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 EquationBasic 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 –Xn
Note here also the definition of = 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 Basic Physics – The Saha EquationEquation
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–I
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 LawBasic Physics – Ideal Gas Law
PV=nRT or P=NkT where N=/
P= pressure (dynes cm-2)V = volume (cm3)N = number of particles per unit volume = density of gm cm-3
n = number of moles of gasR = Rydberg constant (8.314 x 107 erg/mole/K)T = temperature in Kelvink = Boltzman’s constant (1.38 x 10–16 erg/K) = mean molecular weight in AMU (1 AMU =
1.66 x 10-24 gm)
Basic Physics – Thermal Velocity Basic Physics – Thermal Velocity DistributionsDistributions
• 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 IntensitySpecific 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
dAdwdtdv
dEI
cos
dIJ4
1
J is the mean intensity averaged over 4 steradians
FluxFlux• 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
• Luminosity is the total energy radiated from the star, integrated over a full sphere.
• F=Teff4 and L=4R2Teff4
dIF cos 2/
0
cossin2
dIF
Black Black BodiesBodies
• Planck’s Law
• Wien’s Law – Iis maximum at =2.9 x 107/Teff A
• Rayleigh-Jeans Approx. (at long wavelength)
I = 2kTc/ 4
• Wien Approximation – (at short wavelength)
I = 2hc2-5 e (-hc/kT)
1
12/5
2
kThce
hcI
Using Planck’s LawUsing Planck’s Law
Computational form:
For cgs units with wavelength in Angstroms
1
1019.1)(
/1044.1
527
8
Txe
xTB
The Solar NumbersThe Solar Numbers
• F = L/4R2 = 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 Absorption Coefficient and Optical DepthDepth
• 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
dxIdI
L
dx0
eII )0()(
dxd
dIdI
Radiative EquilibriumRadiative 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
dFFxF
00)(
ConvectionConvection• 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 dlog/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.
1log
log
Td
Pd
The Transfer EquationThe Transfer Equation
• For radiation passing through gas, the change in intensity I is equal to:
dI = intensity emitted – intensity absorbed
dI = jdx – Idx
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 Solving the Gray AtmosphereAtmosphere
• Integrating the transfer equation over frequency:
• 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)
SId
dI
cos
TeffT 41))3
2(4
3()(
Monochromatic Absorption Monochromatic Absorption CoefficientCoefficient
• Recall d = dx. We need to calculate , the absorption coefficient per gram of material
• First calculate the atomic absorption coefficient (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 ProcessesPhysical 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 -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 Coeffi cient
0
5E-15
1E-14
1.5E-14
2E-14
2.5E-14
3E-14
3.5E-14
100 600 1200 2200 3200 4200 5200 6200 7200 8200 9200
Wavelength (A)
a (
cm-2
per
ato
m)
x 1
0^
6
n=1
n=2
n=3
PaschenAbsorption
BalmerAbsorption
LymanAbsorption
Neutral hydrogen (bf and ff) is the dominant Source of opacity in stars of B, A, and F spectral type
Opacity from the HOpacity from the H-- Ion Ion
• Only one known bound state for bound-free absorption
• 0.754 eV binding energy• So < 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 Dominant Opacity vs. Spectra TypeType
O B A F G K M
H-Neutral H
H-
Electron scattering(H and He are too highly ionized)
He+ He
Ele
ctr
on
Pre
ssu
r e
High
Low
(high pressure forces more H-)
Low pressure –less H-
The T(The T() Relation) 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:
– 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
SunSun
Star TTeff
TeffT )()(
Hydrostatic EquilibriumHydrostatic Equilibrium
• Since d=dx
•dP/dx= dP/d=gor
dP/d = g/
The Paschen Continuum vs. The Paschen Continuum vs. TemperatureTemperature
Flux Distributions
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
300 400 500 600 700 800 900 1000
Wavelength (nm)
Lo
g F
lux
4000 K
50,000 K
Calculating FCalculating F from V from V
• Best estimate for F at V=0 at 5556A is
F = 3.54 x 10-9 erg s-1 cm-2 A-1
F = 990 photon s-1 cm-2 A-1
F = 3.54 x 10-12 W m-2 A-1
• We can convert V magnitude to F:
Log F= -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 F =-0.400V –8.451 – 0.018(B-V) (erg s-1 cm-2 A-1)
Bolometric CorrectionsBolometric 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
constant5.2 V
bol
F
FBC