The Interior of Stars II

• Stellar Energy Sources (CONTINUED…)

• Energy Transport and Thermodynamics

• Stellar Model Building• The Main Sequence

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

• http://en.wikipedia.org/wiki/Quantum_tunneling

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

•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

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

•Combining all of these for the cross-section

•The reaction rate integral becomes

•Resulting in the “Gamow Peak” at

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

Nuclear Reaction RatesOther 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!!!)

• 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

Nuclear Reaction RatesOther Efffects and Stellar Luminosity

•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 nucleargravity note that gravity can be negative!!!

•Interior Luminosity, Lr, for a spherically symmetric star satisfies:

Stellar Nucleosynthesis and Conservation Laws

• Nucleosynthesis– 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

Proton-Proton Chains

• Proton-Proton Fusion chains

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

p-p chain stage 1

p-p chain stage 2

p-p chain stage 3

p-p chain stage 4

Berrylium Proton-Proton Chains

• PP III (Proton-Proton III Chain)

• 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!!!!

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.

Solar Neutrino Generation

• Too few SNUs!!! (solar neutrino unit)• Solar Model predicts SNU rates with

GREAT CONFIDENCE!!!

• Particle Physics is incomplete!! Neutrino Oscillations

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

Triple Alpha Process of Helium BurningA process to burn Helium…

Very strong temperature dependence!!!!

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

Curve of Binding Energy

• Fusion is an exothermic process until Iron

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

Energy Transport and Thermodynamics

• Radiation

• Convection

• Conduction

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

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 10.6 we obtain

Internal Energy and the First Law of Thermodynamics

• First Law of Thermodynamics

• For an Ideal Gas, the internal energy is:

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

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 10.78 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

Sound Speed

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 10.85 and 10.87 gives the

Adiabatic Temperature Gradient

•Using 10.6 and the ideal gas law we obtain

The Adiabatic Temperature Gradient

•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

•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

A Criterion for Stellar Convection

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

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

•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)

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

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 10.96 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

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