Stars and Planets
—Matter under astrophysical conditions—
Chris Ormel
Roadmap module 2
Mean molecular weight
(for a gas) The relation between mass density and number density
Hydrostatic Balance
The law that connects pressure to gravity
Equation of State (EoS)
derived from the Pressure Integral.
An Equation of State (EoS) describes the relationship between pressure, density and temperature.
The EoS is essential for understanding the interiors of stars (and planets)
Polytropes
Objects with an EoS P~ρ1+1/n obey a self-similar density structure. Many objects in Astronomy (White Dwarfs, neutral stars, some stars, planets) can be described as polytropes
Nuclear Fusion
The energy source that powers stars. Which physical principles underpin this essential energy source? What factors determine the energy production by nuclear fusion?
Stellar Nuclear Synthesis
How are heavy atomic nuclei produced from lighter ones?
- Fermi/Boltzmann distribution
- Ideal Eos
- (rel.)elec.deg. EoS
- radiation Eos
- Mass-radius relationships WD
- Chandrasekhar mass
- Gamow peak
- pp-chain, CNO cycle, Triple-α process
- s-process
- r-process
dominate by mass
of hydrogen is often denoted as X.
Here X=0.75
Here Y=0.23
referred to as "metals" Z. Here Z=0.02
set after the big bang
(cosmic nucleosynthesis).
Metals are produced in stars by nuclear fusion
(stellar nucleosynthesis)
astronomy periodic table
- 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
Blackboard
- Equations of State (dimensionless)
- Pressure Integral
- Electron degeneracy
- Mean molecular weight
mean molecular weight
(read CO Ch.10.2, p288-296)
The Atomic weight of a species per type X is:
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
Here, a particle types
stand for either
ions (
), electrons (
) or simply all particles (
).
With this definition, we can write the number density of particle type X as:
| species | type X | M/mu | NX | μX |
|---|---|---|---|---|
| molecular hydrogen | molecules | 2 | 1 | 2 |
| molecular oxygen | molecules | 32 | 1 | 32 |
| ionized hydrogen | ions | 1 | 1 | 1 |
| ionized hydrogen | electrons | 1 | 1 | 1 |
| ionized hydrogen | all | 1 | 2 | 0.5 |
| ionized helium | ions | 4 | 1 | 4 |
| ionized helium | electrons | 4 | 2 | 2 |
| ionized helium | all | 4 | 3 | 1.33 |
| ionized metal | ions | 2Z | 1 | 2Z |
| ionized metal | electrons | 2Z | Z | 2 |
Common examples
mean molecular weight
If there are multiple species in the mixture, we define the Mean molecular weight μ as:
is the number density of type X of particle species j and
is the weight fraction of the species in the mixture.
where Aj is the mass number and Zj the atomic number of a certain element. It follows that for a fully ionized gas with weight fractions in hydrogen (X), helium (Y) and metals (Z):
Note:
has been inserted. Here, the mass fraction in metals (Z) should not be confused with the atomic number.
| species | type X | M/mu | NX | μX |
|---|---|---|---|---|
| molecular hydrogen | molecules | 2 | 1 | 2 |
| molecular oxygen | molecules | 32 | 1 | 32 |
| ionized hydrogen | ions | 1 | 1 | 1 |
| ionized hydrogen | electrons | 1 | 1 | 1 |
| ionized hydrogen | all | 1 | 2 | 0.5 |
| ionized helium | ions | 4 | 1 | 4 |
| ionized helium | electrons | 4 | 2 | 2 |
| ionized helium | all | 4 | 3 | 1.33 |
| ionized metal | ions | 2Z | 1 | 2Z |
| ionized metal | electrons | 2Z | Z | 2 |
Common examples
- What is μ for Earth's atmosphere?
- What is μ for the Sun?
- What is μe for a White Dwarf?
EOS (Equation of State)
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:
- with mu the atomic mass unit
![]()
- nonrelativistic motion particles
- particles interact only through collisions (no long-range electrostatic forces)
- no internal structure (excitation)
EoS — Degenerate pressure
(read CO Ch16.3 p. 563-569)
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 ne!)
The Fermi momentum is
In the degenerate case, electrons are prevented from filling low-momenta states, providing a much higher pressure than the classical limit for low temperature or high density.
EoS — Degenerate pressure
Fully degenerate electron gas
- non-relativistic, degenerate:
![]()
- relativistic degenerate:
![]()
Non-relativistic and relativistic limits. The term in the [] is the electron number density ne.
Radiation Pressure. We can derive:
The Fermi momentum is
In the degenerate case, electrons are prevented from filling low-momenta states, providing a much higher pressure than the classical limit for low temperature or high density.
EoS (Equation of State)
With these crude expression, we identify several regions:
- radiation pressure at high temperatures.
Stars are unstable in this region (why?)
- ideal gas in the intermediate regions
this holds for stars like the Sun through much of their life
- at low temperatures and high densities pressure is dominated by degenerate electrons
- non-relativistic degeneracy
- relativistic degeneracy
electron degeneracy is important for White Dwarfs and (sometimes) the cores of stars
