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