Stars and Planets

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—Stellar Evolution—

Chris Ormel

Roadmap module 3: Stellar Evolution

Opacity

The "resistance" light experiences in penetrating matter

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Eddington Luminosity

The Luminosity at which radiation outweighs gravity

Heat transport

Mechanism to transport heat (radiative diffusion, convection, conduction)

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Stellar structure equations

→ M2. Hydrostatic Balance

The equations that govern the structure and evolution of stars. Usually solved numerically with codes like MESA

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Homology

A simplification to the stellar structure equations that allows for analytical scaling solutions

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Stellar evolution

→ M1. HR Diagram

A description of how stars interior and appearance changes with time and how these changes depend on parameters, such as the initial mass

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High-mass stellar evolution

High-mass stars deviate from evolution from low-mass stars and play a critical role in nucleosynthesis

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  • radiative and convective temperature gradient
  • Schwarzschield stability criterion
  • mass-radius and mass-luminosity relationships
  • Hayashi track
  • H (shell) burning
  • He flash
  • Stellar nucleosynthesis

Opacity

  • Random walk & mean free path
  • Thomson cross section
  • Eddington Luminosity
  • Opacity sources

Mean free path and opacity

— read CO Ch. 9.2 and 9.3 —

 

Examples of three random walks. The mean free path is 10% of the radius of the circle. Eventually the photon escapes (statistically after 102 random walks).
  • (photon) mean free path : distance before a photon gets absorbed or scattered:

    ni: number density of absorbers per unit volume;
    σ: absorption cross section

  • opacity : inverse mean free path — the transparancy — normalized by density

    Other definitions for opacity exist. In our definition it is the total cross section per unit mass of the medium (units: cm2 g-1).

  • optical depth : number of mean free path length over a length L

Mean free path and opacity

— read CO Ch. 9.2 and 9.3 —

 

the optical depth is depth normalized by the mean free path of the medium.
  • (photon) mean free path : distance before a photon gets absorbed or scattered:

    ni: number density of absorbers per unit volume;
    σ: absorption cross section

  • opacity : inverse mean free path — the transparancy — normalized by density

    Other definitions for opacity exist. In our definition it is the total cross section per unit mass of the medium (units: cm2 g-1).

  • optical depth : number of mean free path length over a length L

Mean free path and opacity

— read CO Ch. 9.2 and 9.3 —

 

only radiation below τ~1 will emerge
  • (photon) mean free path : distance before a photon gets absorbed or scattered:

    ni: number density of absorbers per unit volume;
    σ: absorption cross section

  • opacity : inverse mean free path — the transparancy — normalized by density

    Other definitions for opacity exist. In our definition it is the total cross section per unit mass of the medium (units: cm2 g-1).

  • optical depth : number of mean free path length over a length L

Eddington Luminosity

— read CO 10.6 —

 

Accelerating charges radiate energy at a rate (Larmor formula):

(cgs units) with P is the power, q the charge the acceleration, c the speed of light.

  • The Thomson cross section for photons to interact with electrons

    Valid in the low-energy (no recoil) limit, the photon energy does not change. It follows from equating the radiated power to the incoming energy flux:

    The Thomson cross section can also be obtained by equating the electrostatic energy corresponding to a sphere of radius rT with the rest energy of the electron The corresponding cross section is the Thompson cross section barring numerical constant.

schematic of the interaction of an electromagnetic wave with a charged particle. Due to oscillating E-field, the particle (electron) experiences a force of

in the direction of the E-field, causing it to radiate. In the low-energy limit, the frequency of the radiation is just the same as that of the incoming wave, resulting in a scattering of the EM-wave (Thomson scattering).

Eddington Luminosity

— read CO 10.6 —

 

  • The Thomson cross section for photons to interact with electrons

This results in an (outward) radiation force

  • Balancing with the gravitational force, results in the Eddington Luminosity

    Another, more general, way to write this is, is in terms of the opacity κ — the cross section per unit mass κ=σ/(me+mp) in this case. We then obtain .

The Eddington Luminosity sets an upper mass for stars!

although electrons are point particles, they do interact with photons. The cross section σ over which they interact (in the non-relativistic limit) is known as the Thomson cross section.

opacity

— see CO 9.2 —

 

Sources of opacity:

  • electron scattering

    where is the Thomson cross section

    • Electron scattering dominates in stellar interiors. where gas is ionized.
    • At very high T (frequency), we have that (transition to Compton scattering)

Question: Why can't free electrons absorb photons?

Question: Why the (1+X) factor?

photon scattering and absorption: bound-free, bound-bound, free-free, and electron scattering. In stellar interiors, electron scattering determines the opacity.

opacity

— see CO 9.2 —

 

Sources of opacity:

  • electron scattering
  • Free-free and bound-free

    (cgs units). These forms are Kramer's laws where frequency-dependent opacities have been averaged over the radiation field (Rosseland mean). Expressions are very approximate and are valid only in a certain T-range.

Question: How do we get the combined opacity at a single frequency, κν?

Question: Why is the Rosseland mean opacity harmonically averaged over the radiation field?

bound-free opacities for the H-atom

These opacities are so called Rosseland mean opacities:

that is, the inverse of the opacity (the "transparancy") is weighted by the radiation field.

Why the sudden drop in opacity to 0 at these "cliffs"?

opacity

— see CO 9.2 —

 

Sources of opacity:

  • electron scattering
  • Free-free and bound-free
  • negative H (H—) opacity

    valid for 3,000 < T < 6,000 and -10 < log10 ρ < -5. Important in "cool" atmospheres, where neutral H can bind a second electron. This electron has a low binding energy (compared to neutral H); it dissociates with photons energies above 0.75 eV (~104 K).

  • conductive opacities

    degenerate regions

bound-free opacities for the H-atom
HI bound-free extinction coefficient per hydrogen atom in level n. These must be weighted by the level population and added. (c) Gray (1992).

These opacities are so called Rosseland mean opacities:

that is, the inverse of the opacity (the "transparancy") is weighted by the radiation field.

solar opacities

 

use opacity tables to find as function of temperature and pressure.

We identify with increasing T and ρ:

  1. H— opacity near the Sun's photosphere
  2. strong increase in H— opacity. After this peak H-ionization reduces the H— opacity
  3. bound-free opacities, followed by free-free dominate in the interior
  4. solar core. Opacities just higher than electron scattering limit
a
b
c
d
Radiative opacity for stars, https://opalopacity.llnl.gov/existing.html

solar opacities

 

use opacity tables to find as function of temperature and pressure.

We identify with increasing T and ρ:

  1. H— opacity near the Sun's photosphere
  2. strong increase in H— opacity. After this peak H-ionization reduces the H— opacity
  3. bound-free opacities, followed by free-free dominate in the interior
  4. solar core. Opacities just higher than electron scattering limit
Opacities extracted from MESA opacity table. The opacity plotted is a harmonic mean of the Rosseland-mean opacity and the conductive opacity. The conductive opacity dominates at the bottom-right corner.

Diffusive energy transport

  • Fick's law & Diffusive transport
  • Radiative diffusion
  • Rosseland mean opacity

Radiative heat transport

— read CO 10.4 —

 

Energy transport by diffusion of radiation results in an energy flux (units: [E L-2 T-1]) of

with the radiation energy density, the photon diffusivity, the mean free path, c the speed of light, the opacity. Equating this to the luminosity flux, one obtains

where is the opacity weighted by the radiation field (Rosseland mean). Following convention, the temperature gradient is recast in terms of a gradient with respect to pressure

With the hydrostatic balance equation, the

Radiative temperature gradient becomes

needs to adjust in order for the energy to be transported by radiation!

spread of a dye dissolved in water. Diffusion acts to make the concentration of the uniform, which results in a flux of the dye towards regions of low concentration (down). (c) wikipedia

Energy transport

  • Convective heat transport
  • Radiative and convective gradients

Radiative and convective heat transport

 

The stability condition against convection is

the adiabatic exponent. A strong density gradient (stratification) — dense material below light material — stabilizes fluids.

If we define the EoS as:

where for an ideal EoS. In the adiabatic case, we obtain

the stability criterion transforms to:

mixing length
lighter blobs
rise
deposite energy
after lmix
Convective heat transport: blobs are lighter than the environment, causing them to rise to deposit their energies in cooler layers. Image source: wikipedia

Radiative and convective heat transport

 

some thermodynamical relations may be helpful

where are the heat capacities at constant pressure and temperature. from this the heat capacity ratio follows. Another is the first law of thermodynamics:

if we put we obtain a relationship between the heat capacity ratios

mixing length
lighter blobs
rise
deposite energy
after lmix
Convective heat transport: blobs are lighter than the environment, causing them to rise to deposit their energies in cooler layers. Image source: wikipedia

Radiative and convective heat transport

 

When the radiative gradient becomes too steep, convective "blobs" will rise because they stay lighter than the surroundings. The condition for convection is:

which is Schwarzschield criterion (ignoring any composition gradient). More preciese, in a convective medium, the relation

holds.

  • the actual temperature gradient is less than the radiative gradient as part of the flux is carried by convection.
  • must hold for convection
  • Blobs themselves are never fully adiabatic ( ).

Theories as Mixing length theory offer a phenomenological model to calculate the energy flux that can be carried by convection. It turns out that that under most (stellar) conditions is an acceptable approximation. In other words, convection in stars is very effective. The temperature gradient only needs to be slightly above adiabatic to carry the entire energy flux!

Schematic for the temperature gradient

Radiative and convective heat transport

 

In mixing length theory the convective blobs are assumed to travel for a mixing length before equilibriating with the surroundings

where δq and δT are differences betweeen the heat contents and temperature of the blobs with its surroundings, e.g., cP is the heat capacity at constant pressure, vbl is the velocity of the blobs. After further manipulation and phenomenological reasoning, one obtains

where α ~ β ~ 1 and If we take (the convective bubbles are anyway modelled as adiabatic), it follows that for stellar conditions only a slight overadiabicity, suffices to transport the energy flux by convection.

mixing length
lighter blobs
rise
deposite energy
after lmix
Convective heat transport: blobs are lighter than the environment, causing them to rise to deposit their energies in cooler layers. Image source: wikipedia

Blackboard

  • Stellar structure equations
  • (Photospheric) boundary conditions

Stellar structure equations:

read CO p.330—334

Stellar structure equations

  • Lagrangian formulation is used; radius coordinates r exchanged for mass m
  • the temperature gradient must be provided. Very often

    which is the Schwarzschield criterion. But other choices (e.g. Ledoux criterion) are available.

where is mass, is radius, P pressure, T temperature, l luminosity, density, G Newton's gravitational constant and the temperature gradient, s entropy, the energy production rate, and Xi the composition.