• stars' composition follow cosmic abundances

• enrichment of the ISM (interstellar medium), increase Z with time
  • population III stars (hypothesized) — formed directly after the big bang. No metals.
  • population II stars — old stars. Few metals
  • population I stars — metal-rich. Like our Sun

for planets, the contribution of metals to their composition is much higher

We define the atomic weight for a particle type X as:

where is the atomic mass units ( proton mass). The atomic weight is simply the average mass of the medium per particle type X in units of "Common particle" types are: ions ( ) and electrons ( ). Very often, e.g., if there is only one particle type the index is dropped ( ). Same for the total.
With this definition, we can write the number density of particle type X as:

Common examples

If there are multiple species in the mixture, we define the Mean molecular weight μ as:

here is the number density of type X of particle species j and is the weight fraction of the species in the mixture.

where A_j is the mass number and Z_j the atomic number. It follows that for a fully ionized gas of hydrogen (X), helium (Y) and metals (Z)

Common examples

Pressure integral:

where f(p)dp is the number density of particles with momenta [p,p+dp]

with the Maxwell-Boltzmann distribution

where n is the number density.

This results in the ideal gas law:

Fermi distribution (electrons). For low temperature and/or high density, instead

This follows from the Pauli exclusion principle. Let

be the number of particles in 6D cell Pauli exclusion principle states that there are at most 2 ways to occupy these quantum cells:

Note that in terms of spherical shell momenta d³p = 4πp²dp. The Fermi distribution limits the classical (Boltzmann) distribution (which can become arbitrarily high as it scales with n_e!)

Fully degenerate electron gas

• non-relativistic, degenerate:
• relativistic degenerate:

  Non-relativistic and relativistic limits. The term in the [] is the electron number density n_e.

Radiation Pressure. We can derive:

With these crude expression, we identify several regions:

1. radiation pressure at high temperatures.

   Stars are unstable in this region (why?)

2. ideal gas in the intermediate regions

   this holds for stars like the Sun through much of their life

3. at low temperatures and high densities pressure is dominated by degenerate electrons
   1. non-relativistic degeneracy
   2. relativistic degeneracy

   electron degeneracy is important for White Dwarfs and (sometimes) the cores of stars

hydrostatic balance governs the structure of many astrophysical objects (stars, planets, gas clouds, disks) at rest

where P is pressure, ρ density, g_s the gravitational acceleration along the s-dimension. One may also speak of a hydrostatic (pressure gradient) force, or acceleration g_hs. In vector notation:

mass conservation (in spherical geometry) implies that

where dm is the mass of the mass shell. As an aside, for fluids, the more general mass continuity equation reads:

Often, the equation of state can be approximated as a polytrope

where n is the polytropic index and K a constant.

Hydrostatic balance and mass conservation result in the Lane-Emden equations

where the central density, and the radius has been nondimensionalized as:

Hydrostatic balance and mass conservation result in the Lane-Emden equations

Only n=0, 1, and 5 have analytic solutions.

others are found by numerical integration

numerically, solve it by writing it as a system of first order equations

simple python implementation by your instructor. Here I have used scipy.integrate.odeint to let the computer do the hard work. The only tricky issue is that ψ'(x) is undefined (0/0) for x=0. I strongly encourage you to reproduce these results.

Properties of polytropes

• central-to-average density:

  where ξ_R is the value of ξ where ψ_n becomes zero (outer radius)

• Mass-radius relationship:

  where N_n is a numerical constant.

• Central pressure—density relationship:

  As C_n turns out to be similar for each n, this is a very general relation for the conditions at the center of objects in hydrostatic balance

↑ Proportionality constants appearing in relationships involving mean and central density (ρ_c), mass M, radius R, and central pressure P_c for polytropes for selected polytropic indices n. In particular, note that N and C only weakly depend on n.

Application of polytropes

• For n=3/2 we obtain from the mass-radius relationship: 

  This holds for White Dwarfs (K is a true constant). The more massive the WD, the smaller it is!

• For n=3 the radius dependence vanishes; applied to the K of the ER-degeneracy, we obtain the  Chandrasekhar mass:

  which is the the maximum mass a degenerate object may have. As White Dwarfs have μ_e=2,

The Chandrasekhar mass is the highest mass up to which degenerate objects (White dwarfs) can exist. A star with M>M_Ch must collapse under its own gravity

↑ Proportionality constants appearing in relationships involving mean and central density (ρ_c), mass M, radius R, and central pressure P_c for polytropes for selected polytropic indices n. In particular, note that N and C only weakly depend on n.

Caution!

1. Meaning of K:
   • constant K (e.g., White Dwarfs)

     K is some function of fundamental constant as, e.g., with White Dwarfs. There is a unique mass-radii relationship.

   • variable K (e.g., main-sequence stars)

     Now K is constant for a single star, but not among stars with different mass. All scaling relationships still hold, but there is no unique mass-radius relationship.

     For example, we will see in M4 that for radiatively envelopes, with the proportionality constant determined by luminosity, opacity, etc, that differs among stars. With the ideal gas law, this gives and hence with the prefactor varying among stars. But if is a function of mass only, we can still obtain a mass-radius relationship.

2. The polytropic assumption (cnst n) will fail.
   • White Dwarfs may have relativistically degenerate cores, non.rel.deg interiors, and a thin atmosphere where the ideal EoS applies. n is not constant throughout the star.
   • Material may be in a partial degenerate state.
   • MS-stars may have radiative interiors, convective cores, etc.

← Polytropes are a useful analytical tool, but in reality calculations are done numerically.

Tunneling probability. Temperatures at the center of the Sun (~10⁷ K) are not high enough to overcome the Coulomb barrier. But nuclei can quantum-mechanically tunnel at a probability

(WKB approximation; See any textbook on quantummechanics, e.g., Griffiths.) where p denotes momentum, r_E is the distance corresponding to the Coulomb barrier at given energy E and r_s the distance where the strong nuclear force appears,

is the energy scale where the de Broglie wavelength and Coulomb potential meet, with m_μ is the reduced mass.

Reaction rate — the number of reactions per unit volume per unit time — follows from an n-σ-v argument:

where n₁, n₂ are the number densities of the reactants, v₁₂ the relative velocity, and σ₁₂ the cross section for the reaction. In the second step we have introduced a distribution (and dropped indices).

The nuclear cross section is given by the de Broglie wavelength and the tunneling probability:

where encapsulates all subtleties that come with a more rigorous treatment — like resonances. Essentially, this factor should be measured.

Reaction rate — the number of reactions per unit volume per unit time — follows from an n-σ-v argument:

Inserting for , , and :

where for f(E) we have inserted the Boltzmann energy distribution.

The largest contributions arises from the energies where the argument of the exponent is smallest. This is known as the Gamow peak.

Reaction rate

The largest contributions arises from the energies where the argument of the exponent is smallest. This is known as the Gamow peak.

• Only high-energy nuclei can fuse!
• Nuclear reaction rate very sensitive to temperature!

• H-fusion. The net reaction is
  • This net reaction can be accomplished through the proton-proton (PP) chain: ^(a)
  Notes:
  1. Other variations of the PP-chain exist. This is the dominant pathway for the Sun.
  2. There's an important catch to the first (rate-limiting) reaction, which is a combination of proton fusion and β-decay of one of the protons:

     but the last reaction (β-decay) is very unlikely. In most of the times the diproton just fissions back to two protons. The β-decay is the rate-limiting step

  3. Deuterium burning.

• H-fusion. The net reaction is
  • This net reaction can be accomplished through the proton-proton (PP) chain:
  • Another way to achieve the same net result is the CNO cycle:

    ^† this is the rate-limiting step

• H-fusion. The net reaction is
• He-fusion. In the triple alpha process, He-nuclei fuse into carbon

Nuclear binding energy E_b is energy that is released by building the nucleus from its constituents.

— m_p, m_n: mass of proton, neutron
— A: total number of nucleons (mass number)
— Z: total number of protons (atomic number)
— m_nucl: actual (measured) mass of the nucleus
— c: speed of light

Stars on the main-sequence convert H into He-4, liberating large amount of energy!

After He, heavier elements can be formed if temperatures are high enough:

• (triple-alpha process)
• On to Fe-56 (high mass star)

Heavy elements ("Z") are returned to the ISM by winds during the AGB phase and supernovae explosions:

you are made from stardust

nuclear reaction rates feature a very strong dependence on temperature