1.  The Interior of Stars III
Stellar Model Building
The Main Sequence

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

3. Stellar Model Building Equations of Stellar Structure
Pressure( equation 10.6)
Mass
Luminosity
Temperature
Constitutive Relations

4. Stellar Modeling Difference Equations Numerical modeling shells
Boundary Conditions
Analytic Solutions
For SIMPLIFIED conditions!!!
Cow approximated as a sphere!!!
Difference Equations
Numerical modeling
shells

5. Vogt-Russell Theorem
Pressure Gradient at a given radius is dependent on the interior mass and the density.
Radiative temperature gradient depends on the local temperature density, opacity, and interior luminosity.
Luminosity gradient depends on density and energy generation rate.
Pressure ,opacity and energy generation rate depend explicitly on the density, temperature and composition at that location.
If interior mass at the surface of the star is specified,along with composition,surface radius and luminosity,application of the boundary conditions at surface
P,Mr,T,Lr at a distance dr below the surface
Numerical integration gives the rest P(r),Mr (r),T (r),Lr (r),

6. Vogt-Russell Theorem
“The Mass and the composition structure throughout a star uniquely determine its radius, luminosity and internal structure, as well as its subsequent evolution”

7. Polytropic Models Simplified Assumptions…
A good zeroth order approximation for a cow is a ….sphere!

8. Polytropic Models
Stellar Models in which Pressure depends on density in the form
Under special conditions can find analytic solutions to the a subset of the equations descrbing a stellar model
If only a simple relationship existed between pressure and density. The equations of stellar structure 10.6 and 10.7 could be solved without reference to the energy equations

9. Polytropic Models

10. Polytropic Model…

11. Polytropic Models…
Relates density and radius….

12. Polytropic Models
Solving leads to the profile of density with r.
The pressure profile is obtained from the polytropic equation of state
Assuming the ideal gas law and radiation pressure for constant composition then the temperature profile is obtained

13. Polytropic Models

15. Polytropic Modesl…

16. Polytropic Models

17. The Analytic Solutions to the Lane-Emden Equation

18. Eddington Standard Model

19. Eddington Standard Model…

20. The Main Sequence
Stellar Spectra --> Vast majority of star’s atmospheres are composed primarily from hydrogen
Nuclear burning of hydrogen in its core will cause a “slow” evolution of interior composition.
Most stars with similar initial composition.
The structures of stars should vary smoothly with mass
Low mass stars pp-chain dominates
Higher mass stars CNO cycle dominates
Need at least Msun to generate enough pressure to stabilize star
Very massive stars subject to thermal oscillations

21. The Eddington Luminosity Limit
Stability of very massive stars directly affected by their high luminosities
In the case of radiation pressure dominating near the surface of a massive star we have for the pressure gradient:
Hydrostatic Equilibrium:
Combining and solving for luminosity we obtain the
Eddington Luminosity Limit
This is the maximum luminosity that the star can have and still maintain hydrostatic equilibrium. If this luminosity is exceeded mass loss occurs….

22. Eddington Luminosity limit

23. Models of Main Sequence Stars

24. Computer Modeling….
Star simulation software for your “enjoyment” may be found at the following link..
Simulation results may be found at the following link...

25. Explain This…. http://www. solarphysics. kva

26. The Sun

27. The Sun

28. The Sun The Solar Interior The Solar Atmosphere The Solar Cycle
Theoretical understanding of stellar structure…Now let’s check it…
The Sun is the closest star…Therefore the best studied.

29. Observations of the Sun
High Precision Measurements of properties of Sun
Composition of Sun’s Surface
Luminosity
Effective Temperature
Radius
Magnetic Fields
Rotation Rates
Oscillation Frequencies (vibrations) throughout its interior
Solar Neutrino Production Rate
Allows for rigorous tests of stellar models and our understanding of the physical processes operating within stellar atmospheres and interiors
Sun viewed in the extreme Ultraviolet

30. The Evolutionary History of the Sun
The Sun is classified as a typical main-sequence star of spectral class G2 with a surface composition of X=0.74, Y=0.24 and Z=0.02 (hydrogen,helium and “metals” fractions)
How did the Sun evolve to this point?
The Sun has been converting hydrogen into helium via the p-p chain during most of its lifetime
Composition and structure changes
Stellar Models for the observed composition of the Sun indicate that the Sun should be approximately 4.57x 109 years

31. Present day interior structure of the Sun
Solar Model may be constructed to ascertain features of the interior
This model can be used also to track how the Sun evolves
The mass fraction of hydrogen at the Sun’s center is believed to have started at X=0.71 and has decreased to X=0.34 at the present
The mass fraction for helium at the Sun’s center has increased from Y=0.27 to 0.64
Diffusive settling has actually increased the fraction of hydrogen at the Sun’s surface by approximately 0.03 and the helium fraction has decreased by 0.03 at the surface

32. Present Day Interior Structure of the Sun Composition
The composition of the Sun is no longer homogenous.
Nucleosythesis
Surface Convection
Elemental Diffusion
Composition varies with Radius

33. Energy Production in the Sun
The largest contribution to energy production occurs about 1/10 of the way out from the center of the Sun
This arises from the Mass conservation equation and simple geometry
As you increase radius the volume of a given shell increases as r2
Even though energy production rate per unit mass may decrease with radius the overall production increases until a maximum shell luminosity is reached at about 0.1 Rsun

34. The Present Day Interior of the Sun Temperature and Pressure
Variation of Temperature and Pressure with radius are forced on the solar structure by the following conditions:
Hydrostatic Equilibrium
The Ideal Gas Law
Composition Structure
Boundary conditions at the Surface dictate that both T and P --> 0 there

35. The Present Day Interior of the Sun Mass and Density
Density decreases rapidly with radius
90% of mass of Sun contained within one-half of its radius

36. The Present Day Interior of the Sun Energy Transport Mechanism
How is the Energy produced in the “fusion zone” at the center of the Sun transported out?
Radiative transport dominates out to about 0.7 Rsun.
When the temperature gradient becomes superadiabatic
Convection becomes dominant at between 0.7 Rsun out to near the surface

37. Present Day Interior of the Sun
A model of the Sun has been developed using the Stellar Structure Equations and fundamental physical principles that is complete and reasonable that is consistent with:
Evolutionary timescale
Global Characteristics of the Sun
Mass
Luminosity
Radius
Effective Temperature
Surface Composition
Precise Measurements of Oscillation Frequencies (chapter 14)
Observed Surface Convection Zone
Problem: Abundance of Lithium at surface!!!

38. 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.

39. Solar Neutrino Detection
615,000 kg of cleaning fluid (100,000 gallons)
One isotope of Chlorine could interact with a neutrino of sufficient energy to produce a radioactive isotope of argon with a half life of 35 days
The threshold energy for this reaction is Mev. 77% of the neutrinos above this threshold are from the reaction
about one Argon atom produced every two days!!!!
Chemical analysis used to count argon atoms would measure neutrino flu

40. Solar Neutrino Flux Prediction
John Bachall: Calculated the expected neutrino detection rate for the Davis Chlorine experiment using a model of energy production in the Sun’s interior based on Boron-8 neutrino production in the p-p chain
Why Boron-8? Energy of these neutrinos were above detection threshold of the Chlorine Experiment.
Measured flux about 1/3 of expected flux…What’s going on?

41. Davis Neutrino Flux Measurement

42. Other Neutrino Detectors
Other detectors with sensitivity to the lower energy neutrinos were developed and built. SAGE, Gallex look for the reaction that converts gallium into germanium
Super-Kamiokande looked for Cerenkov light produced when neutrinos scatter electrons
Results from these detectors confirmed the deficit of solar neutrino flux observed by the Davis experiment
What’s Going on?

44. Super-kamiokande

45. Sudbury Neutrino Observatory
Uses heavy water for detector medium…more sensitive

46. Neutrino Mixing

47. The Solar Atmosphere
Photosphere: Segment of star that emits light. Typically defined to be the region down to an optical depth of 2/3.
Chromosphere: In the Sun, a thin layer just above the photosphere that is visually more transparent than the photosphere. The spectrum of the light generated here is dominated by Hwavelength. Temperature of Chromosphere is up to 20,000K.
Transition Region: In the Sun, a region between the Chromosphere and Corona.
Corona: In the Sun, a type of plasma atmosphere that extends millions of kilometers into space. High temperature.

48. The Solar Atmosphere

49. The Photosphere Why does the solar disk appear sharp?
Region where the observed optical photons originate is known as the photosphere
Base of photoshpere is wavelength dependent since opacity is wavelength dependent
Base defined to be 100km below where the optical depth of 500nm light is unity. At this depth 500=23.6 and the Temperature is 9400K
The minimum temperature, 4000K, of the photosphere occurs at its upper edge about 525 km above the 500=1 level. Above this point the temperature starts to rise.
On average the solar flux is emitted from optical depth = 2/3 where the effective temperature is 5777K.
Why does the solar disk appear sharp?

50. The Solar Disk
Sun Radiates primarily as a black-body in the visible and infrared.
Source of opacity is continuous across wavelength
The continuum opacity is due in part to the presence of H- ions in the photosphere. Only 1/107 H-/neautral hydrogen…
Absorption lines are also produced in the photosphere