Equation of state: it’s easiest to treat the interior of the star as a perfect gas.
where n is number density and μ is mean particle mass in units of mH.
Mass fractions and numbers of atoms
X=mass fraction of hydrogen (for the Sun, X=0.70)
Y=mass fraction of helium (for the Sun, X=0.28)
Z=mass fraction of heavier elements, all of which astronomers call ‘metals’ for ease of reference (for the Sun, X=0.02)
Obviously X+Y+Z=1; if it didn’t, there would be something very wrong.
If we assume that the material is fully ionised-
For hydrogen, number of atoms is and number of electrons is .
For helium, number of atoms is and number of electrons is .
For metals, number of atoms is and number of electrons is .
A is the average atomic weight of heavier elements present; each metal atom contributes ~A/2 electrons.
Degeneracy: in a completely degenerate gas all momentum states up to the Fermi momentum are completely filled. In this case, we can no longer use the perfect gas law.
Number density of electrons within a sphere of radius p0 in momentum space at T=0.
The factor of 2 in the integral covers spin states, the 1/h3 covers phase space density.
From kinetic theory
Non-relativistic complete degeneracy
Relativistic complete degeneracy
Non-relativistic degeneracy is important in the degenerate cores of red giants and the interiors of white dwarfs. At high densities, we need to use the relativistic expression; this shows that the white dwarf collapses further to become a neutron star or black hole (there is no longer a stable minimum radius > 0).
Opacity is the cross-section per unit mass and it is a measure of the rate at which energy flows by radiative transfer (the maths was already covered in Astrophysics II). Sources of stellar opacity are-
- Bound-bound absorption (negligible in interiors)
- Bound-free absorption
- Free-free absorption
- Scattering by free electrons
The hydride ion is also important; it has only a single energy at -0.75eV. A lot of absorption is due to the hydride ion.
At low temperature
At intermediate temperature
At high temperature
κ1, κ2, κ3 are constants for stars of a given composition.
Hydrogen burning- PPI chain: nuclear fusion will be covered in more detail in the Nuclear Physics section; here we will only focus on the relevant reactions
PPI chain reactions
- 1H+1H->2D+e++ν +1.44 MeV
- 2D+1H->3He+γ +5.49 MeV
- 3He+3He->4He+1H+1H +12.85 MeV
Overall 4 1H -> 4He
In one cycle, reactions 1 and 2 will occur twice and reaction 3 will occur once.
Total energy released is 26.71 MeV; 0.26 MeV of this is carried away per neutrino and the remaining 26.19 MeV contributes to the luminosity.
We can also write the energy released as 0.007mc2
Timescales for the reactions at characteristic temperature T=3×107K.
- 14×109 yr- reaction 1 is a weak interaction and the bottleneck of the reaction chain; this is the one that sets the lifetime of a hydrogen burning star.
- 6 seconds- deuterium is burned up quickly.
- 106 yr
Obviously the exact values depend on density and mass fractions.
PPII and PPIII chains can occur once 4He and other elements are sufficiently abundant.
3He+4He-> 7Be+γ +1.59 MeV
7Be+e–-> 7Li+ν +0.86 MeV
7Li+1H-> 4He+4He +17.35 MeV
7Be+1H-> 8B+γ +0.14 MeV
8Be-> 4He+4He +18.07 MeV
4He is acting as a catalyst for the conversion of hydrogen to helium. The total energy released is the same in each case, but the energy carried away by the neutrino is different.
PPI, PPII and PPIII operate simultaneously in a star containing sufficient 4He.
The CNO cycle: if a star starts burning helium, heavier elements start to build up. These elements (C, N, O) catalyse the conversion of hydrogen to helium.
12C+1H-> 13N+γ 2nd slowest reaction
14N+1H-> 15O+γ slowest reaction
The timescale of the cycle is determined by the slowest reaction, but the approach to equilibrium is determined by the second slowest reaction.
In equilibrium λ12Cn12C= λ13Cn13C= λ14Nn14N= λ15Nn15N
λ=reaction rate, n=number density
Observational evidence for CNO cycle
- 13C/12C ratio is ~1/5 in some red giants, compared to ~1/90 on Earth.
- Nitrogen-rich stars have been discovered.
CN(O) bi- and tri-cycle: once every 2500 reactions or so, 15N+1H-> 16O+γ occurs. This leads to two other reaction cycles involving fluorine and oxygen; these are the bi- and tri-cycles, and they have an equilibration time of ~1011 yr.
Helium burning: the onset of helium burning is believed to be accompanied by an explosive reaction in an electron degenerate hydrogen-exhausted stellar core. This is the helium flash.
The main problem with helium burning (which we can think of as fusing α-particles) is that the next stable nucleus that can be formed is 12C and after that 16O (no stable nucleus at A=8)- but in order to from these nuclei from α-particles, we would need three or four to collide at the same time, as in the triple- α reaction-
The reaction rate for this is negligible unless on resonance.
There is, however, another possibility. In 4He gas, a small concentration of 8Be can build up and interact with another particle before it decays-
This reaction also has to be resonant to be non-negligible, but a resonant state has been observed.
The characteristic temperature of helium burning is 108 K.
Carbon burning: in large stars, further contraction and heating can occur, allowing fusion of heavier elements all the way up to iron; however, most of the possible energy release from fusion reactions comes from hydrogen and helium burning.
Structure of the Sun: due to its conveniently local position at the heart of our solar system, the Sun is the only star for which we can measure internal properties. Composition can be determined from meteorites; density and internal rotation from helioseismology and central conditions from neutrinos.
Helioseismology: the Sun acts as a resonant cavity, oscillating in millions of mode (acoustic, gravitational). These modes are excited by convective eddies with τ=1.5-20 min, v~0.1 ms-1. Just like earthquakes on Earth, these resonant modes can be used to reconstruct the internal density structure of the Sun. To do this, we need to measure Doppler shifts ~10-6 times smaller than spectral linewidths- this is achieved with good spectrometers and long integration times (to average out noise).
Results of helioseismology
- Density structure and speed of sound in the Sun.
- Depth of outer convective zone is found to be ~0.28RO.
- Core rotation is slow, so it must have been spun down with the envelope.
Solar neutrinos: we already saw that neutrinos are produced in the core of the Sun, and carry away 2-6% of the energy from H-burning. By detecting solar neutrinos in underground experiments, we can probe the solar core.
PP chain neutrino emitting reactions: just to refresh our memories-
1H+1H->2D+e++ν Eνmax=0.42 MeV
7Be+e–-> 7Li+ν Eνmax=0.86 MeV
8B-> 8Be+ν Eνmax=14.0 MeV
The Homestake (Davis) experiment (~1970) to detect neutrinos. An underground tank is filled with 600 tons of C2Cl4. Some neutrinos interact with Cl via the reaction νe+37Cl -> 37Ar+e–. Every two months, 37Ar atoms are filtered out and counted.
There are two problems with this experiment-
- Only a tiny number of neutrinos can be detected.
- It is only sensitive to neutrinos from the 8B reaction, which is only a minor reaction in the Sun.
That aside, ~54 37Ar atoms are expected to be seen, but only 17 37Ar are observed, i.e. the neutrino flux is only about 1/3 of what it should be. This is the solar neutrino problem.
Possible solutions to the solar neutrino problem
- Astrophysical solutions- perhaps our picture of the Sun’s core is wrong. Maybe the central temperature is 5% less than we think, although to achieve this we would need there to be mixing in the core, by convection or rotation.
Other proposed models include there being no nuclear reactions in the Sun (core is black hole/iron/degenerate), or WIMPs (weakly interacting massive particles) transporting away some of the energy in place of the neutrinos. Ultimately, however, helioseismology rules out most astrophysical solutions.
- Nuclear physics- perhaps our reaction cross-sections are wrong. Improved experiments have since confirmed that the cross-sections are correct for the key nuclear reactions.
- Particle physics. All neutrinos generated by the Sun are electron neutrinos, which are the only type the Davis experiment could detect. But if neutrinos have mass, and different neutrino types have different masses, then maybe neutrinos are changing type on the way to earth (neutrino oscillations). This solution looks hopeful, but brings up more considerations- do the oscillations occur in matter or vacuum?
Recent solar neutrino experiments use more sensitive detectors that can also detect neutrinos from the main pp reaction.
- Kamiokande experiment: 3000 tons of ultra-pure water (1680 tons of active medium) for inelastic scattering ν+e– -> ν+e–
This reaction is about six times more likely for νe than νμ or ντ. The observed flux is half the predicted flux, perhaps due to the energy dependence of neutrino interactions.
- Gallium experiments (GALLEX, SAGE) use Ga to directly measure low-energy pp neutrons.
νe+71Ga -> 71Ge+e– -0.23 MeV
Predicted 132±7 SNU
Observed 80±10 SNU
(1 SNU=10-36 interactions per target s-1)
- Sudbury Neutrino Observatory (SNO): 1000 tons of D2O in an acrylic plastic vessel with 9456 light sensors/PMTs, located 2070m underground. The experiment detects Cerenkov radiation from electrons and photons from weak interactions and neutrino-electron scattering.
Results from 2001 seem to confirm neutron oscillations (in matter- MSW effect).
The structure of main sequence (hydrogen core burning) stars: when a star begins its main sequence lifetime, it has a homogeneous composition. During its main sequence lifetime, we can derive certain scaling relations (of the form ). Differential equations are replaced with characteristic quantities in order to derive these.
Hydrostatic equilibrium for main sequence: (1)
Radiative transfer for main sequence: (2)
Luminosity-mass relationship for main sequence: we need to first specify both the equation of state and the opacity law.
- Massive stars: ideal gas law, electron scattering opacity.
- Low-mass stars: ideal gas law, Kramer’s opacity law
- Very massive stars: radiation pressure, electron scattering opacity
Most important thing to remember for stars near a solar mass:
During the main sequence lifetime: core temperature Tc is fixed at ~107 K because that is the characteristic temperature of hydrogen burning. Pressure is inversely proportional to mean molecular mass (μ). During hydrogen burning, μ increases from ~0.62 to ~1.34, whilst the star’s radius increases by a factor of ~2.
Opacity and metallicity: at low temperatures, opacity depends strongly on metallicity ( for bound-free absorption).
Low metallicity stars are much hotter and more luminous at a given mass, hence they have shorter lifetimes.
The mass-radius relationship is only weakly dependent on metallicity.
Subdwarfs are low metallicity stars lying just below the main sequence.
General properties of homogeneous stars
Upper main sequence: M>1.5MO; core is convective and well mixed; energy release via CNO cycle; opacity due to electron scattering; surface H is fully ionised with energy transport by radiation.
Lower main sequence: M<1.5MO; core is radiative; energy release via PP chain; opacity follows Kramer’s Law; surface H/He is neutral with a convective zone just below the surface.
In both cases, temperature increases with mass, and density decreases with mass.
Limits on stellar masses: the lower limit on H-burning is ~0.08MO. Brown dwarfs are low mass objects supported by electron degeneracy. Once the core becomes degenerate hydrogen fusion is never ignited because contraction (and thus heating) ceases.
The upper limit on stars is 100-150 MO.
Note that giants, supergiants and white dwarfs cannot be chemically homogeneous stars supported by nuclear burning as it doesn’t fit in with their properties.