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-
-Determining the Internal Structures of Stars
-Starting in the 1960s computers became advanced enough to carry out calculations for modeling the internal structure of stars.
-The study of stellar structure shows us that stars change over very slow timesclaes compared to human life.
-Some stellar events can be very rapid and dramatic.
-Stars have a limited supply of energy, where they eventually use up the energy and die.
-Stellar evolution is the result of a constant fight against the pull of gravity.
-7.1.2
-The Derivation of the Hydrostatic Equilibrium Equation
-An electrostatic force has opposite charges, where gravitational force is always attractive.
-
-An opposing force must exit in a star to avoid collapse.
-The pressure within a star varies with depth.
-Consider a cylinder of mass dm whose base is located a distance r from the center of a spherical star.
-
-The top and bottom of the cylinder each have an area A and the height is dr.
-We assume that the only forces acting on the cylinder are gravity and the force due to pressure, which is always normal to the top and bottom cylinder surface.
-These forces also vary with the cylinder's distance from the center of the star.
-Using Newton's second law F=ma, this is the net force along the central axis of the cylinder dmd2rdt2=Fg+Fp,top+Fp,bot, where the gravitational force Fg and pressure force on the top of the cylinder F_{p,top} are directed inward, while the pressure force on the bottom Fp,bot of the cylinder is directed upward.
-
-The pressure forces on the curved side of the cylinder will cancel due to symmetry and have been explicitly excluded from the expression.
-The pressure force at the top and bottom of the cylinder are at different sepths, and we can rewrite the equation from Newton's second law
-7.1.3
-The Equation of Mass Conservation
-A relationship for spherical symmetry can be derived using mass, radius, and density.
-This can be done by considering a shell of mass dMr and thickness dr located a distance r from the center of the shell.
-we assume that dr is less than r
-the volume of the shell is 4πr2dr
-Using the relation for density, we can find the mass conservation equation, dMrdr=4πr2p
-This equation describes the mass distribution within a star.
-7.2
-Pressure Equation of State
-Gas has the properties of mass, density, temperature, pressure, and volume that describe how it exist.
-This is the state that the gas is in
-The state represents a macroscopic view of the particle and its interactions.
-This information is necessary to derive a pressure equation of state for the gas.
-The ideal gas law is an example of a pressure equation of state. PV=NkT=nRT
-This ideal gas law relates the gas pressure P, volume V, and Temperature T, to the number of particles N.
-k is the Boltzmann constant, and R is the gas constant.
-For astrophysical problems, we need a pressure equation of state that is more general than the ideal gas law.
-7.2.1
-The Derivation of the Pressure Integral
-The pressure of a gas is the force per area exerted on the walls of its container due to collisions.
-A gas is in a cylinder of length x and cross-sectional area A. The gas is composed of point particles, each of mass m, that interact through perfectly elastic collisions.
-The gas exerts a pressure on one end of the container.
-The impact of an individual particle on the container wall is described through a change in momentum.
-Newton's second and third laws can be applied.
-For a perfectly elastic collision, the angle of incidence equals the angle of reflection.
-The particle approaches the wall in the positive x-direction and rebounds along the negative x-direction.
-Using Newton's 3rd Law, the impulse delivered tot he wall is, 2pxi
-this is using the x-component of the particle's initial momentum px.
-The average force exerted by the particle per unit time can be dertermined by the time interval between collisions.
-the shortest time interval is when the particle traverses the length of the cylinder twice before returning for a second reflection.
-The average force exerted on the wall by a single particle over a time period is given by, F=2pxΔt=pxvxΔx.
-In this equation it is assumed that the force vector is normal to the surface.
-The average force per particle having a momentum p is F(p)=pv3Δx
-It is usually the case that the particles have a range of incident angles.
-this makes the number of particles with momenta between p and p+dp similar to the Maxwell-Boltzmann distribution.
-it is given by the expression Npdp.
-The sum of all the forces exerted by particle collisions is F=13∫NpΔxpvdp
-The pressure exerted on the wall is, P=FA=13∫nppvdp.
-This equation is called the pressure integral, which allows us to compute the pressure given a distribution function.