Stellar Opacity 2u46e

6. Stellar Opacity • Opacity κ appears in energy transport equation. • Represents ability of stellar material to absorb radiation, or 1/(heat conductivity). • Need to consider all microscopic processes that can absorb photons at each frequency ν: – – – –

Bound-bound (bb) absorption Bound-free (bf) absorption Free-free (ff) absorption Electron scattering (es)

• Need to combine them all into a single macroscopic quantity. • Also need to consider thermal (electron) conduction (degenerate white dwarf material).

AS 3003

Stellar Physics

6.1 Opacity and mean free path • Efficiency of conductive processes depends on – density of carriers – energy per carrier – mean free path of carriers.

• For photons of energy E=hν, mean free path:

bound-bound bound-free free-free electron scattering

AS 3003

Stellar Physics

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6.2 Opacity in stellar interiors • Opacity depends on: – Level populations of different ion species (bb,bf) – Ionization balance (bf,ff) – Electron density ne (bf,ff,es)

• LTE is a good approximation inside stars. • Level populations of individual species depend on T (Boltzmann) and on degree of ionization. • Degree of ionization depends on ne and T (Saha). • Electron density ne depends on ρ , T and local composition Xi. If fully ionized: H contributes twice as many e– per unit mass as He.

• So ultimately κν= κν(ρ,T,Xi). • Need to define appropriate mean κ over all ν. AS 3003

Stellar Physics

6.3 Frequency-averaged opacity • How do we weight κν when averaging over ν? • Earlier we defined the flux mean opacity:

• Snag: have to solve the monochromatic equation of radiative transfer throughout the star to determine Fν. • We want a purely local approximation! • Fortunately, in TE: • Hence

Chain rule!

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Stellar Physics

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Rosseland mean opacity - 1 • Rearrange and integrate:

• Define Rosseland mean opacity κR:

• Also:

AS 3003

Stellar Physics

Rosseland mean opacity - 2 • Rearrange to get radiative transport equation:

• i.e. same as before, but now we can calculate κR= κR(ρ,T,Xi) from local conditions alone. • Modern stellar structure calculations use tables of precalculated opacities for different chemical mixtures. • Highly complex! Need to include all species and all bb, bf, ff and es processes. • Opacity Project: data available online.

AS 3003

Stellar Physics

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Useful approximate opacities • Approximations useful in specific ranges of T and ρ for constructing simple stellar models: • Intermediate T: bf and ff processes dominate:

• High T: electron scattering dominates:

NB Check your AS2001 notes!

AS 3003

Stellar Physics

6.4 Diffusion approximation • Gradient in energy density of carriers: C=Heat capacity

• Heat flux density is then:

• For photons:

Thermal diffusivity

Mean speed

Mean free path

• For electrons:

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Stellar Physics

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6.5 Electron conduction • e– - ion collisions transfer energy more efficiently than e– - e– collisions. • Mean free path for e– to collide with ion is:

• where r is the distance at which potential energy of e– - ion pair matches thermal KE:

From diffusion approximation AS 3003

Stellar Physics

Electron vs photon conduction • Use electron scattering cross-section to get a lower limit on photon conductivity:

• e.g. hydrogen plasma, solar interior conditions:

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Electron conduction in white dwarfs • Photon mean free path becomes short in dense material. • Electrons form dense degenerate gas with high conductivity, cf.metals. • If Fermi energy εF >> kT:

• Mean free path is also longer: e– only scattered if unoccupied state available to be filled. • High electron conductivity gives nearly uniform temperature throughout degenerate interior. • Surrounded by insulating outer layer of nondegenerate e– and ions. AS 3003

Stellar Physics

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