Stellar structure equations Lecture 9 Stellar structure equations

Review: The Eddington approximation This is the Eddington-Barbier relation: the surface flux is determined by the value of the source function at a vertical optical depth of 2/3

The Eddington Approximation Approximation #5: Local thermodynamic equilibrium In this case the source function is equal to the blackbody function.

The Eddington Approximation Approximation #5: Local thermodynamic equilibrium In this case the source function is equal to the blackbody function. Recall: So

The Eddington Approximation Note that T=Te when tv=2/3 Thus the photons we see (that give us the luminosity we use to define the effective temperature) originate at an optical depth of 2/3, not 0. Te

Limb darkening The solar disk is darker at the edge (limb) than at the centre The light rays that we see from the edge of the Sun must originate from higher in the atmosphere (since otherwise they would have to travel through a greater optical depth to reach us).

Limb darkening Now we will again assume a plane-parallel atmosphere, but we will not assume a gray atmosphere, LTE, or the Eddington approximation We need to find the intensity as a function of angle, q Recall the definition of the vertical optical depth: So we can rewrite the general solution as:

Limb darkening We can compare this with observations, and the agreement is pretty good. This doesn’t prove our numerous assumptions are correct, but does show that they do not produce a result that is in strong conflict with the data.

Break

Hydrostatic equilibrium The force of gravity is always directed toward the centre of the star. Why does it not collapse? The opposing force is the gas pressure. As the star collapses, the pressure increases, pushing the gas back out. How must pressure vary with depth to remain in equilibrium?

Hydrostatic equilibrium Consider a small cylinder at distance r from the centre of a spherical star. Pressure acts on both the top and bottom of the cylinder. By symmetry the pressure on the sides cancels out dr A dm FP,b FP,t

Hydrostatic equilibrium If we now assume the gas is static, the acceleration must be zero. This gives us the equation of hydrostatic equilibrium (HSE). It is the pressure gradient that supports the star against gravity The derivative is always negative. Pressure must get stronger toward the centre

Mass Conservation The second fundamental equation of stellar structure is a simple one relating the enclosed mass to the density. Consider a shell of mass dMr and thickness dr, in a spherically symmetric star Rearranging we get the equation of mass conservation

Example Imagine a sphere of gas with density profile What is the total mass and average density within radius r?

Example Make a crude estimate of the central solar pressure, assuming the density is constant. This is a big underestimate because the density increases strongly toward the centre. The accepted value is

Pressure equation of state We now need to assume something about the source of pressure in the star. We require an equation of state to relate the pressure to macroscopic properties of the gas (i.e. temperature and density) Consider the ideal gas approximation: Gas is composed of point particles, each of mass m, that interact only through perfectly elastic collisions Note the particle mass does not enter into this equation The momentum of each collision depends on mass, but lighter particles are moving faster in a way that exactly cancels out Thus, tiny electrons contribute as much to the pressure as massive protons

Mean molecular weight We want to relate the particle number density to the mass density of the gas. The two quantities are related by the average particle mass: Define the mean molecular weight: i.e. this is the average mass of a free particle, in units of the mass of hydrogen So we can express the ideal gas law in terms of density:

Mean molecular weight The mean molecular weight is an important quantity, because the pressure support against gravity depends on the number of free particles Sudden changes in the ionization state or chemical composition of the star can lead to sudden changes in the pressure. In general, the value of m requires solving the Saha equation to determine the ionization state of every atom. We can derive two useful expressions for the cases of fully neutral or fully ionized gasses: Define: X,Y,Z are the mass fractions of H, He and metals, respectively. Neutral: Ionized: For solar abundances,

Example By how much does the pressure increase following complete ionization, for a neutral gas with the following composition (typical of young stars):

Radiation pressure The equation of state then becomes We earlier derived an expression for the radiation pressure of a blackbody: The equation of state then becomes In a standard solar model, the central density and temperature are Calculate the gas pressure and radiation pressure. Assume complete ionization, so m=0.62.

Non-ideal gases The ideal gas law neglects both special relativity and quantum mechanics. It therefore breaks down at high velocities (temperatures) and at high densities. The Fermi-Dirac distribution is a modification of the Maxwell-Boltzmann distribution, accounting for the Heisenberg uncertainty principle and the Pauli exclusion principle Essentially, no two fermions (e.g. electrons and protons) can occupy the same quantum state. This provides an extra source of pressure when densities get high

Non-ideal gases The ideal gas law neglects both special relativity and quantum mechanics. It therefore breaks down at high velocities (temperatures) and at high densities. The Bose-Einstein distribution function applies to bosons (such as photons). In this case, the presence of a particle in a quantum state enhances the probability that another particle will occupy the same state Both the Fermi-Dirac and Bose-Einstein distribution functions approach the Maxwell-Boltzmann distribution at high energies