Lecture 12 Stellar structure equations. Convection A bubble of gas that is lower density than its surroundings will rise buoyantly  From the ideal gas.

Slides:



Advertisements
Similar presentations
Nuclear Astrophysics Lecture 5 Thurs. Nov. 21, 2011 Prof. Shawn Bishop, Office 2013, Ex
Advertisements

1 The structure and evolution of stars Lecture 3: The equations of stellar structure Dr. Stephen Smartt Department of Physics and Astronomy
1 The structure and evolution of stars Lecture 2: The equations of stellar structure Dr. Stephen Smartt Department of Physics and Astronomy
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit.
Thermodynamics versus Statistical Mechanics
Nuclear Astrophysics Lecture 4 Thurs. Nov. 11, 2011 Prof. Shawn Bishop, Office 2013, Extension
Prof. Shawn Bishop, Office 2013,
Solar interior Solar interior Standard solar model
Nuclear Astrophysics 1 Lecture 3 Thurs. Nov. 3, 2011 Prof. Shawn Bishop, Office 2013, Ex
First Law of Thermodynamics
The structure and evolution of stars
Lecture 5 First Law of Thermodynamics. You can’t get something for nothing. Nothing is for free. We will discuss these statements later…
1 The structure and evolution of stars Lecture 7: The structure of main- sequence stars: homologous stellar models.
Stellar Interior. Solar Facts Radius: –R  = 7  10 5 km = 109 R E Mass : –M  = 2  kg –M  = 333,000 M E Density: –   = 1.4 g/cm 3 –(water is.
Mechanical equivalent of heat Joule (1843) Under adiabatic conditions 1 °F increase when 772 lb dropped 1 foot J = 1 cal 1 J ≡ amount of work required.
Stellar Interiors Astronomy 315 Professor Lee Carkner Lecture 10.
First Law of Thermodynamics Physics 313 Professor Lee Carkner Lecture 8.
Properties of stars during hydrogen burning Hydrogen burning is first major hydrostatic burning phase of a star: Hydrostatic equilibrium: a fluid element.
Heat Physics 313 Professor Lee Carkner Lecture 9.
Dr. Jie ZouPHY Chapter 20 Heat and the First Law of Thermodynamics (cont.)
Stellar Structure Section 3: Energy Balance Lecture 4 – Energy transport processes Why does radiation dominate? Simple derivation of transport equation.
Lecture 6 Adiabatic Processes. Definition Process is adiabatic if there is no exchange of heat between system and environment, i.e., dq = 0 dq = 0.
The First Law of Thermodynamics
Stellar Structure Section 3: Energy Balance Lecture 5 – Where do time derivatives matter? (part 1)Time-dependent energy equation Adiabatic changes.
Stellar Structure Section 4: Structure of Stars Lecture 7 – Stellar stability Convective instability Derivation of instability criterion … … in terms of.
Stellar Structure Chapter 10. Stellar Structure We know external properties of a star L, M, R, T eff, (X,Y,Z) Apply basic physical principles From this,
Hydrostatic Equilibrium Physical Astronomy Professor Lee Carkner Lecture 9.
Properties of stars during hydrogen burning Hydrogen burning is first major hydrostatic burning phase of a star: Hydrostatic equilibrium: a fluid element.
Interesting News… Regulus Age: a few hundred million years Mass: 3.5 solar masses Rotation Period:
Lecture 10 Energy production. Summary We have now established three important equations: Hydrostatic equilibrium: Mass conservation: Equation of state:
The Interior of Stars II
Stellar Structure Gas Mass Radiation Energy Generation Transport Radiative Convective Temperature Density Composition Hydrostatic Equilibrium:
Nuclear Astrophysics Lecture 9 Thurs. Dec. 22, 2011 Prof. Shawn Bishop, Office 2013, Ex
Review of Lecture 4 Forms of the radiative transfer equation Conditions of radiative equilibrium Gray atmospheres –Eddington Approximation Limb darkening.
The Interior of Stars I Overview Hydrostatic Equilibrium
Stellar structure equations
1 The structure and evolution of stars Lecture 5: The equations of stellar structure.
ATOC 4720: class The first law of thermodynamics 1. The first law of thermodynamics 2. Joule’s law 2. Joule’s law 3. Specific heats 3. Specific heats.
1 The structure and evolution of stars Lecture 4: The equations of stellar structure.
P203/4c17:1 Chapter 17: The First Law of Thermodynamics Thermodynamic Systems Interact with surroundings Heat exchange Q = heat added to the system(watch.
The Sun and other stars. The physics of stars A star begins simply as a roughly spherical ball of (mostly) hydrogen gas, responding only to gravity and.
1B11 Foundations of Astronomy Sun (and stellar) Models Silvia Zane, Liz Puchnarewicz
1 Flux Transport by Convection in Late-Type Stars (Mihalas 7.3) Schwarzschild Criterion Mixing Length Theory Convective Flux in Cool Star.
Lecture 11 Energy transport. Review: Nuclear energy If each reaction releases an energy  the amount of energy released per unit mass is just The sum.
METR February Review Hydrostatic balance Pressure decreases exponentially with height, isothermal atmosphere: Zeroth law of thermodynamics:
Chapter 4: Applications of the First Law Different types of work: Configuration work: (reversible process) Dissipative work: (irreversible process) Adiabatic.
The Sun. Discussion What does it mean to say the Sun is in hydrostatic equilibrium?
1 The structure and evolution of stars Lecture 3: The equations of stellar structure.
Lecture 15 main sequence evolution. Recall: Initial cloud collapse A collapsing molecular cloud starts off simply:  In free-fall, assuming the pressure.
PHYS377: A six week marathon through the firmament by Orsola De Marco Office: E7A 316 Phone: Week 1.5, April 26-29,
Heat & The First Law of Thermodynamics
Conservation. Work Expanded  Mechanical work involves a force acting through a distance.  Work can involve a change in internal energy. Temperature.
Static Stellar Structure. 2 Evolutionary Changes are generally slow and can usually be handled in a quasistationary manner Evolutionary Changes are generally.
Q18. First Law of Thermodynamics. 1.A quantity of an ideal gas is compressed to half its initial volume. The process may be adiabatic, isothermal or isobaric.
Lecture 29: 1st Law of Thermodynamics
Hydrodynamics Continuity equation Notation: Lagrangian derivative
Lecture 8: Stellar Atmosphere 4. Stellar structure equations.
On expanding isothermally from 2L to 4L, an ideal gas does 6J of work, as the pressure drops from 2 atm to 1 atm. By how much must it expand to do an additional.
The First Law of Thermodynamics
Outline – Stellar Evolution
Quasistatic processes The relation of heat and work
Thermodynamics Universe Surroundings System Heat Work Mass
Modeling Stars.
Thermodynamics Universe Surroundings System Heat Work Mass
Chapter 4 Energy Balances without reaction.
Flux Transport by Convection in Late-Type Stars (Hubeny & Mihalas 16
Convection John Crooke 3/26/2019.
Conservation.
The structure and evolution of stars
Presentation transcript:

Lecture 12 Stellar structure equations

Convection A bubble of gas that is lower density than its surroundings will rise buoyantly  From the ideal gas law: if gas is in approximate pressure equilibrium (i.e. not expanding or contracting) then pockets of gas that are hotter than their surroundings will also be less dense.

Convection Convection is a very complex process for which we don’t yet have a good theoretical model

The first law of thermodynamics For an ideal, monatomic gas:

The first law of thermodynamics In a stellar partial ionization zone, where some of the heat is being used to ionize the gas. In isothermal gas For an adiabatic process (dQ=0): From the ideal gas law for ideal, monatomic gas

Polytropes For an adiabatic, monatomic ideal gas For radiative equilibrium, or degenerate matter For isothermal gas A polytrope is a gas that is described by the equation of state:

Convection Assume that the bubble rises in pressure equilibrium with the surroundings. What temperature gradient is required to support convection? Using the ideal gas law and the equation for hydrostatic equilibrium:

Convection Compare the temperature gradient due to radiation: with that required for convection: Simulation of convection at solar surface Observations of granulation on solar surface When will convection dominate?

Static Stellar structure equations Hydrostatic equilibrium: Mass conservation: Equation of state: Energy generation: Radiation Convection Polytrope or

Break

Derivation of the Lane-Emden equation 1. Start with the equation of hydrostatic equilibrium 2. Substitute the equation of mass conservation: 3. Now assume a polytropic equation of state: 4. Make the variable substitution:

The Lane-Emden equation So we have arrived at a fairly simple differential equation for the density structure of a star: This equation has an analytic solution for n=0, 1 and 5. This corresponds to  =∞, 2 and 1.2 n=0,1,2,3,4,5 (left to right)

Stellar structure equations For the polytropic solution, we can easily find the temperature gradient, using the ideal gas law and polytropic equation of state. This is equal to the adiabatic temperature gradient: Finally, to determine the luminosity of the star we use the equation Where the energy generation  depends on density, temperature and chemical composition.

Thermodynamics Convection is the transport of heat: thus we need to understand the basic laws of thermodynamics. The first law of thermodynamics: energy conservation Change in internal energy Heat added Work done on the surroundings For a mass element dm: The internal energy U is a state function: it depends only on the current state of the gas (not the processes leading up to that state)  The change dU is independent of the process for that change, whereas dQ and dW are not.