1. The Interior of Stars I Overview Hydrostatic Equilibrium
Pressure Equation of State
Stellar Energy Sources
Next lecture
Energy Transport and Thermodynamics
Stellar Model Building
The Main Sequence

2. The Interior of Stars I Calculate This!!! Use Knowledge of:
Thermodynamics
Properties of light and how it interacts with matter
Nuclear Fusion
Basic Parameters of Star
M: Mass
L: Luminosity
Te: Effective Surface Temperature
R: Radius

3. Hydrostatic Equilibrium

4. Thermal Equilibrium

5. Opacity

6. Energy Transport: Radiative Transport

7. Energy Transport: Convection

8. Energy Generation: Thermonuclear Fusion
Binding Energy of Nuclei can be released in the form of Energy (photons,…)

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

10. Hydrostatic Equilibrium
Let’s determine the internal structure of stars!!!
Some guidance:
Hydrostatic Equilibrium: Balance between gravitational attraction and outward pressure
Gravity
Pressure Gradient
Net Force on Cylinder

11. Derivation of Hydrostatic Equilibrium
Substituting 10.2 and 10.3 into 10.1
Density of Gas Cylinder
Gives
Dividing by volume of cylinder
If star is static, we then obtain:
Pressure Gradient
for hydrostatic equilibrium

13. The Equation of Mass Conservation
Relationship between mass, density and radius
Mass of shell at distance r
Where  is the local density of the gas at radius r.
Rearranging we obtain

14. Pressure Equation of State
Where does the pressure “come from”? How is it described?
Equation of State relates pressure to other fundamental parameters of the material
Example: Ideal Gas Law
Derivation of the Pressure Integral for a cylinder of gas of length x and area A
Newton’s 2nd law
Impulse delivered to wall
Average force exerted on wall by a single particle
What is the distribution of particle momenta?
The average force per particle is then
If the number of particles with momenta between p and p+dp is Npdp. Then the total number of particles in the cylinder is
Contribution to the total force by all particles in the momentum range p and p+dp is

15. Pressure Equation of state The Ideal Gas Law in Terms of the Mean Molecular Weight
Integrating over all possible values of momenta the total Force is:
Dividing both sides by the surface area of the wall A gives the pressure P=F/A. Noting that V=Ax and defining npdp to be the number of particles per unit volume
We find that the pressure exerted on the wall is:
Recast in terms of velocities for non-relativistic particles with p=mv
In the case of an Ideal Gas the velocity distribution is given by the Maxwell-Boltzmann distribution
Particle number density is
Substituting into Pressure integral we obtain
Pressure Integral
Given the distribution function npdp. The pressure can be computed

16. Pressure Equation of state The Ideal Gas Law in Terms of the Mean Molecular Weight
Expressing particle number density in terms of mass density and mean particle mass
The Ideal gas law becomes
Mean Molecular Weight
Re-expressing in terms of mean molecular weight

17. Mean Molecular Weight
The mean molecular weight depends on the composition of the gas as well as the state of ionization for each species. For completely neutral of completely ionized the calculation simplifies.
For Completely neutral
Dividing by mH
For completely ionized gases, we have
Where (1+zj) accounts for the nucleus plus the number of free electrons that result from completely ionizing an atom of type j

18. Mean Molecular Weight Re-expressing using that for a neutral gas
Thus for a neutral gas

19. Mean Molecular Weight

20. The Average Kinetic Energy Per Particle
Combining and 10.9 we see that
This can be re-written as:
For the maxwell-boltzmann distribution
Hence the average kinetic energy per particle is
3 from 3 degrees of freedom from 3-d space

21. Maxwell-Boltzmann Statistics
Classical distribution of energy of particles in thermal equilibrium

22. Fermi-Dirac Statistics
Particles of half-integral spin are known as Fermions and satisfy fermi-dirac statistics
Some Fermions: electrons,protons,neutrons
Influences Pressure….

23. Bose-Einstein Statistics
Particles of integral spin are known as Bosons and satisfy Bose-Einstein statistics
Photons are Bosons
Influences Pressure….

24. The Contributions due to Radiation Pressure
Because photons possess momentum they can generate a pressure on other particles during absorption or reflection
The Pressure integral can be generalized to photons
In terms of energy density
For a blackbody distribution one has
Total Pressure=Gas Pressure+Radiation Pressure

26. Stellar Energy Sources Gravitation and the Kelvin-Helmholtz Timescale
One likely source of stellar energy is gravitational potential energy.
Graviational potential energy between two particles is
Gravitational force on a point particle dmi located outside of a spherically symmetric mass Mr is:
The potential energy is then
Consider a shell with
Where  is the mass density…Thus
Integrating over all mass shells from the center to the surface

27. Gravitation and the Kelvin-Helmholtz Timescale

28. Gravitation and the Kelvin-Helmholtz Timescale

29. Energy Generation: Thermonuclear Fusion
Binding Energy of Nuclei can be released in the form of Energy (photons,…)

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

31. The Nuclear Timescale The binding energy of He nucleus is
This energy can be released thru a process in which 4 protons are combined into a He nucleus through the process known as Fusion. This particular reaction can occur through several processes …p-p chain, CNO cycle,….
How much energy is available in a star from this fusion process?

32. Nuclear Timescale