Spring 2016

__TOPIC LIST AND READING ASSIGMENTS__

*Physical introduction to stars:*Mass conservation, gravitational contraction, free-fall, hydrostatic equilibrium (HSE), virial theorem, pressure and kinetic energy density for non-relativistic and ultra-relativistic gases, equilibria, star formation, Jeans mass, Jeans density, contraction of protostar, conditions for stardom (Phillips sections 1.1 to 1.4)*Stellar fluxes and luminosities:*distances, parallax, flux, photon luminosity, definition of magnitude scale, apparent and absolute magnitudes, distance modulus, bolometric corrections, stellar photometric systems, neutrino luminosity, mass-loss luminosity (Clayton pp. 1-14, Phillips pp. 33-34)*Temperature, thermal equilibrium, and statistical mechanics:*definition of temperature, Maxwell-Boltzman, Fermi-Dirac, and Einstein-Bose statistics, occupation indices, number of states available to a free particle, Blackbody (BB) radiation, Planck function, Stefan-Boltzmann law (Clayton pp. 14-22; Phillips pp. 49-51)*Stellar temperatures:*(Clayton pp 22-36)- Effective temperature (temperature of BB with same radiated power per unit area)
- Color temperature (temperature of BB with the same spectral shape): opacity of stellar atmospheres, limb darkening, color indices
- Excitation temperature (from relative strengths of stellar absorption lines from a single ion or atom): Boltzmann formula, statistical weights, Einstein coefficients
- Ionization temperature (from relative strengths of stellar absorption lines from two different ionization states of the same atom): Saha equation, partition function (also Phillips pp. 66-78)

*Stellar classification:*History from Fraunhofer to WISE, including Williamina Fleming, Annie Jump Cannon, and Antonia Maury; temperature sequence (OBAFGKM, plus LTY); luminosity classes; Wolf-Rayet stars, carbon stars; white dwarf classification; MLTY: from stars to brown dwarfs (see powerpoint slides on sakai; Clayton pp. 36-39)*Other relevant stellar observables:*(Clayton pp. 39-41)- Mass from applying Kepler's laws to binary stars (orbital motion must be measured), best measured mass is for the sun: 1 solar mass = 1.989e33 gm.
- Radius from interferometry (Betelgeuse was the first to be measured [in 1920], current data puts its size at approximately 1000 solar radii), lunar occultation, eclipsing binaries (distance must be known), best measured radius is for the sun: 1 solar radius = 6.957e10 cm (IAU-defined nominal solar radius)
- Composition from analysis of spectral features

*Hertzsprung-Russell diagram:*(Phillips pp. 33-37; Clayton pp. 46-51)*Ages of stars:*luminosity is a strong function of mass (L propto M^3; Phillips Fig. 1.4); stellar age is roughly proportional to the amount of fuel (M) divided by the rate of fuel use (L), so stellar age is also a strong function of mass (t propto M^2)*Star clusters:*open clusters, globular clusters, color-magnitude diagrams, stellar evolution, model stellar evolution tracks, age and distance determination, stellar populations defined from higher to lower metallicity as Pop I, Pop II, and Pop III (see powerpoint slides on sakai; Phillips pp. 37-38; Clayton pp. 51-62)*Stellar energetics:*age of sun from Kelvin-Helmholtz time (~30 million years) is too short; nuclear fusion now known to be the source of the sun's luminosity, Schonberg-Chandrasekhar limit (Clayton pp. 41-46, Phillips pp. 23-28)*Thermodynamic state of the stellar interior:*develop understanding of how pressure depends on composition, density, and temperature, perfect (ideal) gases are a good approximation to completely ionized gas up to very high temperatures (Clayton pp. 77-79; also see Phillips chapter 2 for much of the following)- Mechanical pressure of a perfect monatomic non-degenerate gas, mean mass per particle, X=mass fraction of hydrogen, Y=mass fraction of He, Z=mass fraction of everything else so Z=1-X-Y (Clayton pp. 79-86)
- Pressure of degenerate electrons, quantum concentration, non-relativistic complete degeneracy, relativistic complete degeneracy, partial degeneracy (Clayton pp. 86-99)
- Density-Temperature equation of state regimes for an ideal electron gas (Phillips Fig. 2.2)
- Pressure from ions: usually only a small additional term to the electron pressure for degenerate cases.
- Radiation pressure (Clayton pp. 105-112)
- Quasistatic changes of state, review of basic thermodynamics, derivation of ratio of specific heats for ideal monatomic, diatomic, polyatomic gases, add radiation, derive three adiabatic indices (Clayton pp.112-123)
- Effects of ionization, calculate the first adiabatic index for varying degrees of hydrogen ionization (Clayton pp.123-129)
- Estimate internal pressure of sun from HSE, uniform contraction/expansion (Clayton pp. 130-134)
- Derive virial theorem from equations of motion of mutually interacting particles (Clayton pp. 130-134)

*Polytropes:*idealized stellar models with a polytopic equation of state that satisfies mass conservation and equation of HSE (no heat tranport or energy generation), express pressure as a power law of density for different cases with different exponents of the polytropic index, "n", plausible examples include Kelvin's "boiling star" (star in adiabatic convective equilibrium), Eddington's "standard model" (star in radiative equilibrium) and white dwarfs (more later), Lane-Emden equation, properties of the standard model (Clayton pp. 155-165; Phillips pp. 148-149)*White dwarfs:*endpoint of stellar evolution for the sun (and all stars less massive the 8 solar masses), supported by electron degeneracy, most common are the carbon-oxygen white dwarfs, there are some helium white dwarfs and oxygen-neon-magnesium white dwarfs, mass-radius relation, Chandrasekhar limit (Phillips pp. 171-180)*Energy transport in the stellar interior:*relate rate of energy production in stellar interior to luminosity profile for static stellar structure (Clayton pp. 166-169; Phillips eqn. 3.26; WARNING!!: Clayton and Phillips use different definitions for the same symbol, epsilon, representing the energy production rate: Phillips uses the power generated per unit volume while Clayton uses the power generated per unit mass. Hence Clayton's eqn multiplies his epsilon by rho, the gas density)- Radiative transfer, derive radiative heat flow equation, show role of absorption, scattering, and emission (induced and spontaneous), scattering phase function, Rosseland mean opacity, Chandrasekhar's luminosity formula for stars in radiative equilibrium (Clayton pp. 170-185, Phillips pp. 87-93)
- Opacity, basic review of mean free path, optical depth, cross section, and opacity, four broad categories for the sources of stellar opacity: bound-bound absorption, bound-free absorption (photo-ionization, inverse of radiative recombination), free-free absorption (inverse bremsstrahlung), and free-free scattering (e.g., Thomson scatering, Compton scattering), Thomson scattering, Kramers opacity law (Clayton pp. 185-186, 199-201, eqn. 3-151, pp. 213-216, eqn. 3-157, pp. 222-224, Fig. 3-16)
- Convection, actual blobs of gas rising and falling as they carry heat energy, critical condition for convection, discuss situations in stellar interiors where energy transport is radiative and where it is convective (Clayton pp. 252-256, Phillips pp. 93-95)

*Energy generation in stars:*basic review of the main sequence, H-burning by the proton-proton and CNO cycle (Phillips pp. 23-24, Fig. 4.4)- Relevant physics: fusion of light elements is exothermic, sufficient energy is available, problem is overcoming the Coulomb barrier (electrostatic repulsion of two like charged nuclei), nuclear are non-degenerate (except in neutron stars) so velocity distribution is Maxwellian, fusion is a balance between the failing high velocity tail of the Maxwellian and the rising probability for quantum mechanical penetration of the Coulomb barrier
- Non-resonant reaction rates (cross section is the product of three parts: an intrinsically nuclear part, the de Broglie wavelength of the particles squared, and the penetration factor), barrier penetration, Gamov peak (Phillips pp. 107-117; Clayton pp. 288-309)
- Temperature dependencies of p-p chain and CNO cycle (Phillips pp. 117-123)
- Equilibrium abundance of deuterium in the p-p chain, CNO abundances in the core during CNO cycle
- Solar neutrinos, Ray Davis' experiment in Homestake gold mine (started in 1968 ran for 30 yr) found ~1/3 expected number of photons based on Bahcall's Standard Solar Model, Kamiokande II confirmed that neutrinos came from the sun but saw only 1/2 the expected number, Sudbury Neutrino Observatory was sensitive to all types of neutrinos (electron, muon and tau), total neutrino flux was consistent with the Standard Solar Model, but only ~1/3 were of the electron type (Phillips pp. 123-127)
- Helium burning, triple alpha, Holye's prediction of a resonant enhancement in the fusion rate for an excited level of 12C (Phillips pp. 127-136)
- Advanced stages of nuclear burning (Phillips pp. 136-139, Norbert Langer's notes)

*Homology relations:*analytic scaling relations that approximate numerical solutions to stellar structure, define homology, applications to luminosity-mass relation and the main sequence*Schematic stellar evolution:*use homology relations to see how stars evolve in the density-pressure and density-temperature equation of state planes*Star formation:*approach to the main sequence, Hayashi track (HT), polytropic model for HT

**Weekly HW Assignments will be posted on the Assignments tab of the sakai course web site and must be turned in through sakai.
Unless otherwise noted, H/W is due by 5:00PM, Friday, of the week indicated. **

Week of |
Topics Covered |
HW Assignment |

Jan 18 | Physical introduction to stars | None |

Jan 25 | Stellar fluxes and luminosities, temperature, thermal equilibrium, and statistical mechanics | Assignment 1 (on sakai) |

Feb 1 | Stellar temperatures, Stellar classification Other relevant stellar observables | Assignment 2 (on sakai) |

Feb 8 | Hertzsprung-Russell diagram, Stellar ages Star clusters, Stellar energetics, LIGO result on GW150914 and stellar endpoints | Assignment 3 (on sakai) |

Feb 15 | Pressure of ideal non-degenerate gas; mean mass per particle, Pressure of ideal degenerate electron gas, density-temperature EoS regimes, Radiation pressure, brief summary of ASTRO-H ("Hitomi") science after successful launch | Assignment 4 (on sakai) |

Feb 22 | Class cancelled Tues (Feb 23); Quasistatic changes, Specific Heats, Adiabatic indices, Effects of ionization | None |

Mar 14 | Spring Break | None |

Mar 21 | Midterm exam (Tues, Mar 22) | Assignment TBD |

May TBD | Final Exam | ... |

Please send any comments to Jack Hughes, jph //at\\ physics.rutgers.edu.

Revised January 25, 2015