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

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

Hydrostatic Equilibrium

Thermal Equilibrium

Opacity

Energy Transport: Radiative Transport

Energy Transport: Convection

Energy Generation: Thermonuclear Fusion

Binding Energy of Nuclei can be released in the form of Energy (photons,…)

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

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

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 Gradientfor hydrostatic equilibrium

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

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

Pressure Equation of stateThe 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=Ax and defining npdp to be the number of particles per unit volume

• We find that the pressure exerted on the wall is:

Pressure Integral

Given the distribution function npdp. The pressure can be computed

•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 Equation of stateThe 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

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

Mean Molecular Weight

• Re-expressing using that for a neutral gas

•Thus for a neutral gas

Mean Molecular Weight

• http://astro.wsu.edu/models/calc/XYZ.html

The Average Kinetic Energy Per Particle

• Combining 10.10 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

Maxwell-Boltzmann Statistics

• Classical distribution of energy of particles in thermal equilibrium

Fermi-Dirac Statistics

• Particles of half-integral spin are known as Fermions and satisfy fermi-dirac statistics

• Some Fermions: electrons,protons,neutrons

• Influences Pressure….

http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics

Bose-Einstein Statistics• Particles of integral spin

are known as Bosons and satisfy Bose-Einstein statistics

• Photons are Bosons

• Influences Pressure….

http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statisticshttp://demonstrations.wolfram.com/BoseEinsteinFermiDiracAndMaxwellBoltzmannStatistics/

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

http://hyperphysics.phy-astr.gsu.edu/hbase/starlog/staradpre.html#c1

Stellar Energy SourcesGravitation 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

Integrating over all mass shells from the

center to the surface

Where is the mass density…Thus

Gravitation and the Kelvin-Helmholtz Timescale

Gravitation and the Kelvin-Helmholtz Timescale

Energy Generation: Thermonuclear Fusion

Binding Energy of Nuclei can be released in the form of Energy (photons,…)

Curve of Binding Energy

Fusion is an exothermic process until Iron

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?

Nuclear Timescale