• (photon) mean free path : distance before a photon gets absorbed or scattered:

  n_i: 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

• (photon) mean free path : distance before a photon gets absorbed or scattered:

  n_i: 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

• (photon) mean free path : distance before a photon gets absorbed or scattered:

  n_i: 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

Sources of opacity:

• electron scattering

  where is the Thomson cross section (see module 4).

  • 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?

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, κ_ν?

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 suddon drop in opacity to 0 at these "cliffs"?

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

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

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, this radiative temperature gradient becomes

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

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:

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!

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 m is mass, r is radius, P pressure, T temperature, l luminosity, ρ density, G Newton's gravitational constant the temperature gradient, s entropy, ε the energy production rate, and X_i the composition.

The stellar structure equations represent a system of ordinary differential equations (ODEs). To solve them, we need boundary conditions:

• at the center, m=0: L=r=0 (T and P unknown)
• at the surface, m=M, we can take zero boundary conditions for temperature and pressure: T=P=0
• better to take photospheric boundary conditions:

  These relations follow from the Eddington approximation for the atmosphere:

  at τ=2/3 — the photosphere — the temperature equals the effective temperature and the pressure can be found by integrating dP/dr to infinity

MESA is a publicly available collection of codes ("modules"), aimed towards 1D stellar evolution, which are contributed by many users from the entire planet!

• it is one dimension (1D), so relatively fast
• illuminating and rich
• large international community

Above all, it's fun! Strongly recommend the ambitious student to download and check it out!

Some insight into the stellar structure equations can be obtained by comparing two stars of different mass, assuming homology

where is the homologous mass coordinate

Note: The conditions for homology are only rarely satisfied.

For example, when a stars' core become degenerate while the outer regions expands — the RGB stage — it breaks

Applying homology to the stellar structure equations, we obtain:

Homology relations obtained from the continuity equation, pressure balance, the (ideal) equation of state, and energy transport (by radiation) equation. LHS should be evaluated at the homologous position x . The latter relation shows that (for constant opacity and composition) , which is approximately the Main Sequence.

Applying the homologous relationship for the center ( ) to the general version of the EoS:

Therefore, for a collapsing star:

Contracting (cooling!) stars heat up for an ideal gas, but contract for a degenerate stars!

Stellar evolution can be understood as a continuous (quasi-static) collapse, interceded by prolonged periods of "steady" nuclear burning

• low-mass stars do not reach H-burning temperatures

  They end up as Brown Dwarfs.

Stellar evolution can be understood as a continuous (quasi-static) collapse, interceded by prolonged periods of "steady" nuclear burning

• low-mass stars do not reach H-burning temperatures
• solar-mass stars reach H-burning and He-burning temperatures, but cannot avoid degenerate cooling once their supply runs out

  Note: after H-burning the temperature in the now He-core would increase due to the higher μ. This is not reflected in the figure.

  Stars end up as degenerate objects (White Dwarfs)

Stellar evolution can be understood as a continuous (quasi-static) collapse, interceded by prolonged periods of "steady" nuclear burning

• low-mass stars do not reach H-burning temperatures
• solar-mass stars reach H-burning and He-burning temperatures, but cannot avoid degenerate cooling once their supply runs out
• high-mass stars avoid the "degeneracy trap", burn many elements

  More massive than the Chandrasekhar mass; They eventually explode as supernovae

Stellar evolution can be understood as a continuous (quasi-static) collapse, interceded by prolonged periods of "steady" nuclear burning

• low-mass stars do not reach H-burning temperatures
• solar-mass stars reach H-burning and He-burning temperatures, but cannot avoid degenerate cooling once their supply runs out
• high-mass stars avoid the "degeneracy trap", burn many elements
• Even higher mass stars would be unstable due to radiation pressure

• In the limit of fully convective stars we can derive the following scaling relationships
  This line is known as the Hayashi line. These relationships follow from the following assumptions:

  • an n=3/2 polytropic (interior) model, , and .
  • an atmospheric model that matches the polytropic model (pressure equilibrium)
  • an opacity relation (H^— opacity)  , so and .
  • The Stefan-Boltzmann relationship (to eliminate R in liu of L)

• The extremely weak dependence on T renders the line vertical in the HR-diagram

  (although more realistic models will feature some structure)

• Stars in hydrostatic balance cannot exist in hydrostatic equilibrium right of this line (at lower T) !

  This is known as the "forbidden zone"

→ Pre main-sequence evolution tracks and observations in Taurus-Auriga star formation complex (Stahler 1988).

The main sequence (MS)

— read CO 13.2 —

• The place in the HR-diagram where stars burn ("fuse") hydrogen
• Stars follow a mass-luminosity relationship with for low-mass MS-stars

  This means high-mass stars quickly leave the MS, while low-mass stars are still on the MS
  The main-sequence turnoff is a way to determine the age of globular cluster stars.

• For stellar clusters, the MS-turnoff is an indicator of its age

Perryman et al. (1997) — Hertzsprung-Russell diagram for stars in the solar neighborhood, which distances have been determined by the Hipparcos spacecraft.

The main sequence (MS)

• The place in the HR-diagram where stars burn ("fuse") hydrogen
• Stars follow a mass-luminosity relationship with for low-mass MS-stars

  This means high-mass stars quickly leave the MS, while low-mass stars are still on the MS
  The main-sequence turnoff is a way to determine the age of globular cluster stars.

• For stellar clusters, the MS-turnoff is an indicator of its age

Question: What does the branch (A) below the MS represent?

Gaia Collaboration et al. (2018) — 20 years later, there is Gaia.

• The place in the HR-diagram where stars burn ("fuse") hydrogen
• Stars follow a mass-luminosity relationship with for low-mass MS-stars
• For stellar clusters, the MS-turnoff is an indicator of its age

• Stars will ascend the Red Giant Branch and (later) the Asymptotic Giant Branch

  The evolution depends on stellar mass and also metallicity

• The final outcome depends mainly on the (initial) stellar mass:

Question: These helium white dwarfs haven't been observed yet. Why not?

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

(B) hydrogen is exhausted in the center; onset of H-shell burning; growth of envelope.

The H-shell burning and the contraction of the (near-)degenerate He core, cause the envelope to expand!

The underlying reason is that nuclear reaction are very sensitive to temperature; a tiny increase in T (due to contraction of the shell), greatly increases the luminosity output

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

Q: Why does the convective zone grow during the RGB phase

(C) the expanding envelope becomes convective; the star moves up ("ascend") the Red Giant Branch (RGB).

The He-degenerate core contracts and grows; Luminosity and nuclear burning increase, resulting in higher mass (and smaller!) core; the high temperature gradient ensures that the nuclear burning shell is thin

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

(D-F) an accelerating evolution along the red giant branch.

The envelope has by now become largely convective (high luminosity and high opacity!); evolution along the Hayashi line

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

Q: Why doesn't the energy liberated in the He flash come out?

(F) He is (explosively) ignited in the degenerate core

A Helium flash occurs; the He-burning is (for a short period) out of control, reaching power levels comparable to the galaxy. However, most of the generated energy does not reach the surface. Because the degenerate core's pressure is insensitive to temperature. We do not observe this as the energy goes into internal energy, lifting the degeneracy!

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

(G-H) He core burning

He-core burning expands the core and causes the envelope to shrink. These He-burning stars occupy the so-called Horizontal branch

Left: Example of a so-called Kippenhahn diagram — evolution of the internal physical structure with time. Red-shaded regions denote nuclear burning, grey convection. Right: corresponding curves in the HR-diagram. (c) Onno Pols

(H-J) He shell burning; the star ascends the Asymptotic Giant Branch (AGB)

On the AGB the star may experience strong pulsations, sheds its outer envelope to the ISM, before its He-fuel is exhausted. A degenerate C/O-core remains as a White Dwarf

• In high-mass stars, a progression of burning cycles results in an onion-shell structure
• Since Fe-burning would consume energy, the core collapses and heats up
• Photons become energetic enough to disintegrate the Fe-group heavy nuclei (photo-disintegration)

  This reverses the nuclear fusion in a few seconds! For example:

• Under the high density and temperatures, protons and electrons combine (electron capture)

  this releases a huge amount of neutrinos (cooling)

• Collapse is halted by degenerate pressure from neutrons core-collapse (Type II) supernovae