1. The Interior of Stars II
Stellar Energy Sources (CONTINUED…)
Energy Transport and Thermodynamics
Stellar Model Building
The Main Sequence

2. Coulomb Repulsion and Quantum Tunneling
Range of strong nuclear force is approximately 1fm=10-15m
Coulomb repulsion barrier until d=1fm…Vc=3.43 Mev between protons
Classically the kinetic energy of proton must exceed this barrier potential
Quantum mechanical tunneling effectively lowers coulomb barrier.
Proton must approach approximately within one Debroglie wavelength of the target

3. Coulomb Repulsion and Quantum Tunneling
Classically proton would need to climb full coulomb repulsion barrier to get close enough (~1fm) for strong nuclear potential to become effective.
Quantum Mechanical barrier penetration allows the proton to tunnel “close enough”
Lowers temperature at which fusion occurs

4. Nuclear Reaction Rates and the Gamow Peak
Nuclear reaction rates depend upon:
Distribution of kinetic energies for a given temperature
Number density in given energy range
Probability for tunneling at that energy
Must integrate over all energies to get total rate
Maxwell-Boltzmann distribution gives the number density in the energy range E to E+dE
Cross-section: Number of reactions per target nucleus per unit time
Number of incident particles contained in the cylinder
Number of incident particles/volume
Number of reactions per target nucleus /time interval dt having energies between E and E+dE is

5. Nuclear Reaction Rates and the Gamow Peak
Total number of reactions per unit volume per unit time
Need to know form of (E). Use idea of DeBroglie Wavelength
Cross-section also proportional to tunneling probability
Combining all of these for the cross-section
The reaction rate integral becomes
Resulting in the “Gamow Peak” at

6. Nuclear Reaction Rates Other Efffects
Resonance
If energy of incoming particle “matches” differences in energy levelsof target nucleus…cross-section goes through a narrow resonance peak
Electron Screening
Ionized electrons produce a “sea of negative charge” that screens electric charge of nuclei effectively reducing Coulomb barrier (10% to 50% enhancement in helium rates!!!)

7. Nuclear Reaction Rates Other Efffects and Stellar Luminosity
Power Laws
Reaction rates can be represented as a power law such as
Can then formulate the amount of energy liberated per kilogram of material per second as (in units of W kg-1)
Can then show the dependence of energy production on temperature and density
The Luminosity gradient equation
To determine the luminosity of a star must consider the energy generated
The contribution to the luminosity from an infinitesimal mass can be expressed as:
Where  is the total energy released per kilogram per second by all nuclear reactions and gravity nucleargravity note that gravity can be negative!!!
Interior Luminosity, Lr, for a spherically symmetric star satisfies:

8. Stellar Nucleosynthesis and Conservation Laws
What are the exact sequence of steps by which one element is converted into another?
Conservation Laws of physics must be satisfied
Electric charge
Number of Nucleons
Number of leptons
…Antimatter

9. Proton-Proton Chains
Proton-Proton Fusion chains

10. Proton-Proton Chain I Each step has its own rate
Slowest step involves the decay of a proton into a neutron via the Weak nuclear force

11. p-p chain stage 1

12. p-p chain stage 2

13. p-p chain stage 3

14. p-p chain stage 4

15. Berrylium Proton-Proton Chains
PP II (Proton-Proton II Chain)
Possible reactions of He-3 and He-4 That produce Beryllium
Via Lithium or Boron
Neutrino Energy Distribution is distinct from PP I chain…
Observations of Solar Neutrino Energy Distributions/rates
Tests
Solar Model
Basic Particle Physics!!!!
PP III (Proton-Proton III Chain)

16. Solar Neutrino Generation
Neutrinos allow the interior of the sun to be viewed “directly” but…
Solar Neutrino Problem too few solar neutrinos observed. Standard solar model predicted greater neutrino flux that that was observed…
Resolution:…Neutrino Oscillations. Neutrinos change “flavor” and become undetectable on their flight from the Sun.

17. Solar Neutrino Generation
Too few SNUs!!! (solar neutrino unit)
Solar Model predicts SNU rates with GREAT CONFIDENCE!!!
Particle Physics is incomplete!! Neutrino Oscillations

18. CNO Cycle CNO energy production Strong temperature dependence
If star begins to collapse density and temperature increase…This results in higher energy production
Collapse balanced by energy production

19. Triple Alpha Process of Helium Burning
A process to burn Helium…
Very strong temperature dependence!!!!

20. Carbon and Oxygen Burning
For massive stars that can generate sufficiently high temperatures and densities …Carbon and Oxygen can be “burned…”
T>6 x 108 K --> C burning
T>109 K --> O burning

21. Curve of Binding Energy
Fusion is an exothermic process until Iron

22. Overview: Equations of Stellar Structure
Pressure
Mass
Luminosity
Temperature
HYDROSTATIC EQUILIBRIUM
GEOMETRY/ DEFINITION OF DENSITY
NUCLEAR PHYSICS
THERMODYNAMICS
(ENERGY TRANSPORT)

23. Energy Transport and Thermodynamics
Radiation
Convection
Conduction

24. The Radiative Temperature Gradient
Radiation Pressure Gradient from ch 9
Equating gives temperature gradient
Using
We obtain an expression of the temperature gradient for Radiative transport

25. The Pressure Scale Height
If temperature gradient becomes too steep, convection can play an important role in energy transport
Convection involves mass motions:hot parcels move upward as denser parcels sink
Characteristic length scale for convection, Pressure Scale Height typically about the size of the star.
To estimate the size of the convective region, consider
We can solve for P(r)
Using equation we obtain

27. Internal Energy and the First Law of Thermodynamics
For an Ideal Gas, the internal energy is:

28. Specific Heats
The change in heat of a mass element for a change in temperature is expressed as the specific heat.
Depends on whether pressure or volume is held constant…
Play with differentials …
Consider a gas in a cyclinder with a piston that moves, we have for dW
Consider
With dV=0, we have
For an ideal monatomic gas
To find Cp for a monatomic gas
Ratio of specific heats

29. The Adiabatic Gas Law The Adiabatic Gas Law
For an adiabatic process (dQ=0) for which no heat flows into or out of the mass element. The first law can be written as:
From with constant n
Since dU=CVdT, we have
Combining
Using (10.79) and (10.80) gives
Solving this differential equation we obtain
The Adiabatic Gas Law
Using the Ideal Gas Law a second adiabatic relation may be obtained

30. Sound Speed

31. The Adiabatic Temperature Gradient
Consider a hot convective bubble of gas that rises and expands adiabatically
After it has traveled some distance it thermalizes, giving up excess heat
Loses its distinction and dissolves into the surrounding gas
Differentiating the ideal gas law (10.11) yields an expression involving the bubble’s temperature gradient (How the buble’s temperature changes with position)
Using the adiabatic relationship between temperature and density (10.82) and specific volume V=1/we have
Differentiating and re-writing, we obtain
Assuming constant combining and gives the
Adiabatic Temperature Gradient
Using 10.6 and the ideal gas law we obtain

32. The Adiabatic Temperature Gradient
If |dT/dr|act is just slightly greater than |dT/dr|ad deep in the interior of the star than this may be sufficient to carry nearly all of the luminosity by convection!!!!
It is often the case that either radiation or convection dominates the energy transport deep in the interior of stars
Dominant Energy Transport mechanism is determined by the temperature gradient in the deep interior
Near surface of star situation is more complicated and both transport mechanisms can be important
This expression describes how the temperature of the gas inside the bubble changes as the bubble rises and expands adiabatically
If the star’s actual temperature gradient is steeper than the adiabatic temperature gradient
The temperature gradient in that case would be said to be superadiabatic

33. A Criterion for Stellar Convection

34. A Criterion for Stellar Convection
What conditions must be met for convection to dominate over radiation in the deep interior of a star?
When will a hot bubble of gas continue to rise rather than sink back down after being displaced upward
Archimedes Principle: If the initial density of the bubble is less than its surroundings it will begin to rise
Buoyant force per unit volume exerted on a bubble immersed in a fluid of density (s) is:
Graviational force on bubble
Net force per unit volume on bubble
If after having traveled dr the density of the bubble becomes greater than the surrounding medium it will sink again and convection is prohibited. Otherwise if the density of the bubble remains less it will continue to rise and convection will result

35. Criteria for stellar convection
Express this condition in terms of temperature gradients. Assume that initially the gas is very nearly in thermal equilibrium and the densities were almost the same
Bubble expands adiabatically
This condition must be satisfied for convection to occur

36. The Mixing Length Theory of Superadiabatic Convection
Pressure of bubble and surrounding gas are always equal
Ideal gas law then implies that bubble must be hotter than surrounding gas
Therfore the temperature of the surrounding gas must decrease more rapidly with radius
Is required for convection to occur. Since the temperature gradients are negative
Assuming that the bubble moves adiabatically , let
After a bubble travels dr, its temperature will exceed the surrounding gas by:
Assume that a bubble has traveled
Before dissipating. At which point it etherealizes giving up its excess heat at constant pressure. The distance l is called the mixing length. Hp is the pressure scale height and
Is a free paramter that is the ratio of the mixing length to pressure scale height. Generally assumed to be about 1. (0.5 -3)

37. The Mixing Length Theory of Superadiabatic Convection
After the bubble travels one mixing length , the excess heat flow per unit volume from the bubble to its surrounding is
Use with dr=l . Multiplying the average velocity of the convective bubble one obtains the convective flux
Average velocity is obtained from the net force per unit volume acting on the bubble
Finally we obtain:
After further manipulations …an expression for the convective flux

38. The Mixing Length Theory of Superadiabatic Convection
To evaluate the convective flux you need to know the difference between the temperature gradients of the bubble and its surroundings…
For simplicity assume that all the flux is carried by convection
This will allow us to estimate the difference in the temperature gradients
Comparing to the adiabatic temperature gradient