Stellar Atmospheres II

• Stellar Opacity..cont’d

• Radiative Transfer

• The Transfer Equation

• The Profiles of Spectral Lines

Optical Depth

http://spiff.rit.edu/classes/phys440/lectures/optd/optd.html

Sources of Opacity

• Bound-Bound Transitions• Bound-Free Absorption (photo-

ionization)

• Free-Free Absorption

• Electron-Electron Scattering

•Continuum Opacity and the H- ion (wavelength < 1640 nm)

Graduate level treatment

http://spiff.rit.edu/classes/phys440/lectures/opacity/opacity.html

Bound-Bound Transitions

• Photons captured by atoms Hydrogen atom transitions

Cross section for photon absorption

Bound-Free Transitions

• Photon Ionizes atoms

Free-Free Transitions

Electron scattering

• Thomson Scattering Cross-section

Balmer Jump

Continuum Opacity and the H- ion

• Hydrogen atoms can “catch” an additional electron

• Binding energy is 0.754 eV

• Photons with wavelength of 1640nm or less can ionize this Hydrogen ion

An additional source of opacity for stars cooler than type F0

Total Opacity is sum of all contributions:

Rosseland Mean Opacity

• An attempt at estimating the average opacity over all wavelengths

• Weight by the rate at which Intensity distribution (blackbody radiation) varies with temperature.

• Determine dependence of other parameters such as temperature

• http://www-star.st-and.ac.uk/~kw25/teaching/stars/STRUC7.pdf

Opacity

• Total Mean Opacity is mean of the sum of the opacity from the various sources

– Bound-bound

– Bound-free

– Free -free

– Electron scattering

– H- ion

• Kramers-opacity law

– Bound-bound

– Bound-free

How far do we look into a star?

• Looking into a star at any angle, we always look back to an optical depth of about as measured straight back along the line of sight

• Photon’s at a distance of less than 1 mean free path from the surface are likely to escape

• Star’s photosphere is defined to be the layer from which visible light originates

• Formation of spectral lines (absorption) occur because temperature of the material in the star decreases outwards from the center of the star

Limb Darkening

• When looking at the center of the sun one can see “deeper” than when looking at the edge of the sun

• Deeper is hotter

• Hotter is brighter…

LIMB DARKENING

What’s this stuff about…?

• Understand:

– Stellar atmosphere.

• P(r),T(r),rho(r)

– Radiation Field

• I(r)

• Opacity,Optical Depth

• Limb Darkening

– Spectrum from Star

– Elemental Abundances

Radiative Transfer

• Energy Transfer in the form of Electromagnetic radiation

• In an equilibrium steady-state star there can be no change in the total energy contained within any layer of the stellar interior or atmosphere

• Implies absorption must be balanced by emission

• Inverse emission processes• Emission processes re-direct photons

from initial path

Random Walk

• http://en.wikipedia.org/wiki/Radiative_transfer

Tortuous path:

100 steps --> d=10 l

10000 steps --> d=100 l

1000000 steps --> d=1000 l

Optical Depth is roughly the number of photon mean free paths from that point to the surface.

Radiation Pressure Gradient

• Mean free path of photons in the interior of the star is typically a few cm.

• Center of star is few hundreds of millions of meters from its surface

• How do the photons ever get out?

• Temperature of star decreases outward from center. This causes the radiation pressure to decrease outwards as well

Pressure Gradient

Photon “Breeze” from star

This causes a slow upward diffusion of randomly walking photons

Pressure Gradient produces a slight net movement of photons upward that carries the radiative flux

The Transfer Equation

• The Emission Coefficient j– Describes the rate of

emission processes

• Source of radiation S=j/

– Ratio of rates of emission to absorption

• Equation of Radiative Transport

• http://astro.u-strasbg.fr/~koppen/apindex.html#STARS

Describes the passage of light through a Star’s Atmosphere

The Transfer EquationCase of Blackbody Radiation

• Particle and Photons in equilibrium individually and with each other

• Every process of absorption is balanced by an inverse process of emission

• Intensity of Radiation described by the Planck Function I = B

• Intensity constant throughout box

• Thermodynamic Equilibrium

Source Function is equal to the Planck function

Plane-Parallel Atmosphere

• Far from center of star the radius of curvature is large…consider plane-parallel (flat) slabs…

• Work in units of optical depth instead of distance…

• Z-axis is vertical z=0 is at top of slab

• Vertical Optical Depth

• Optical depth for inclined ray

• Results in…Transfer Equation

Plane Parallel Atmosphere

• Further simplify by using mean opacity--->”Gray Atmosphere”. No more dependence on wavelength. Replace Iand S.

• We obtain…

• Integrating over solid angles…S doesn’t depend on angles

• And using 9.8 and 4pi.

• We obtain

•Multiplying 9.40 by cosand integrating

•The term on the left is the radiation pressure multiplied by c. The first term on the right is the radiative flux the second integral is zero.Thus…

•Which is just the radiation pressure gradient 9.31. Can be interpreted as the net radiative flux is driven by differences in radiation pressure. This equation will be used in chapter 10 to determine the tempearture structure in the interior of a star.

Plane Parallel Atmosphere

• In an equilibrium stellar atmosphere, every process of absorption is balanced by an inverse process of emission, no net energy is subtracted from or added to the radiation field.

• This means that in a plane-parallel atmosphere, the radiative flux must have the same value at every level in the atmosphere, including its surface….

• Because the flux is a constant

• Can integrate 9.42

To obtain the radiation pressure as a function of optical depth

Eddington Approximation

• Approximate I with Iout and Iin

• Evaluate Temperature as a function of optical depth

Limb Darkening Revisited

Eddington Approximation of Solar Limb Darkening

Profiles of Spectral Lines

• Spectral Line conveys information about the environment in which it was formed

• Line width

• Line Shape– Broadening – Profile

Optically Thin Spectral Line. Thus termed because there is now wavelength at which the Radiant flux has been completely blocked.

Sources of Broadening

• Natural Broadening– From Uncertainty Principle– ~2-4 x 10-5 nm

• Doppler Broadening– From thermal motion of atoms– ~4.27 x 10-2 nm at 5772K

• Pressure and Collisional Broadening– From atomic orbitals being perturbed from collisions– ~2-4 x 10-5 nm

Natural Broadening

More Involved calculation…

Doppler Broadening

More in depth analysis…

Tails in M-B dist Turbulent blobs…

Pressure Broadening

The Voigt Profile

• Combination of all broadening effects

Curve of Growth

Tool to determine the abundances of elements in stellar atmospheres.

• Width of spectral line depends on the number of absorbing atoms

Can determine Abundances of elements!!!!

• Consider an element not present in stellar atmosphere --> No line

• Add some atoms of that element ---> optically thin line

• Double concentration -->width doubles

• Add more -->saturation

• Add even more --> wings of line grows

• Add even more ---> pressure broadening…

Curve of Growth

Tool to determine the abundances of elements in stellar atmospheres.

• Width of spectral line depends on the number of absorbing atoms

Can determine Abundances of elements!!!!

Curve of Growth for the Sun

• Let’s determine the amount of sodium in the Sun!!!

• Observe width of spectral line

• Read off Na from curve of growth

Computer Modeling of Stellar Atmospheres

• Each atmospheric layer is involved in the formation of line profiles and contributes to the spectrum observed for the star.

• Many “ingredients”…

– Opacity

– Radiation Pressure

– Transfer equations

– +

• Hydrostatic Equilibrium

• Statistical and Quantum Mechanics

• Transport of energy by radiation and convection….

Calculate:• how temperature, pressure and density vary with depth.•Line profiles•Effective Temperature•Surface Gravity,….

Worked Problems

Worked Problems

Worked Problems

Worked Problems

Worked Problems

Worked Problems