The Interior of Stars III

• Stellar Model Building• The Main Sequence

Overview: Equations of Stellar Structure

• Pressure

• Mass

• Luminosity

• Temperature

• http://abyss.uoregon.edu/~js/ast121/lectures/lec22.html

THERMODYNAMICS(ENERGY

TRANSPORT)

NUCLEAR PHYSICS

GEOMETRY/ DEFINITION OF DENSITY

HYDROSTATIC EQUILIBRIUM

Stellar Model Building Equations of Stellar Structure

• Pressure( equation 10.6)

• Mass

• Luminosity

• Temperature

• http://abyss.uoregon.edu/~js/ast121/lectures/lec22.html

Constitutive Relations

Stellar Modeling

• Boundary Conditions

• Analytic Solutions – For SIMPLIFIED

conditions!!!– Cow approximated as a

sphere!!!

Difference Equations•Numerical modeling•shells

• 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),

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”

Vogt-Russell Theorem

Polytropic ModelsSimplified Assumptions…

A good zeroth order approximation for a cow is a ….sphere!

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

Polytropic Models

Polytropic Model…

Polytropic Models…

Relates density and radius….

Polytropic Models

• Solving 10.110 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

Polytropic Models

Polytropic Modesl…

Polytropic Models

The Analytic Solutions to the Lane-Emden Equation

Eddington Standard Model

Eddington Standard Model…

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 0.072 Msun to generate enough pressure to stabilize star

• Very massive stars subject to thermal oscillations

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….

Eddington Luminosity limit

Models of Main Sequence Stars

Computer Modeling….

Star simulation software for your “enjoyment” may be found at the following link..

http://www.physics.utah.edu/~springer/phys3060/Assignments_files/ModernAstrophysicsCode

Simulation results may be found at the following link...

http://www.physics.utah.edu/~springer/phys3060/Assignments_files/AppendixL

Explain This….http://www.solarphysics.kva.se/gallery/movies/oslo-2004/movies/cont6302_20Aug2004_sunspot.mpg

The Sun

The Sun

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.

Observations of the Sun

Sun viewed in the extreme Ultraviolethttp://umbra.nascom.nasa.gov/eit/

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

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

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

Present Day Interior Structure of the SunComposition

• The composition of the Sun is no longer homogenous.

– Nucleosythesis

– Surface Convection

– Elemental Diffusion

• Composition varies with Radius

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

The Present Day Interior of the SunTemperature 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

The Present Day Interior of the SunMass and Density

• Density decreases rapidly with radius• 90% of mass of Sun contained within

one-half of its radius

The Present Day Interior of the SunEnergy 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

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!!!

Solar Neutrino GenerationNeutrinos 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.

• 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 0.814 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

Solar Neutrino Detection

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?

Davis Neutrino Flux Measurement

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?

Super-kamiokande

Sudbury Neutrino Observatory

• http://www.sno.phy.queensu.ca/

• Uses heavy water for detector medium…more sensitive

Neutrino Mixing

http://en.wikipedia.org/wiki/Neutrino_oscillation#Solar_neutrino_oscillation

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 Hwavelength. 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.

The Solar Atmosphere

The Photosphere

• 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?

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