lec-19-20 - Lectures 1920: White Dwarfs and Neutron Stars...

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Unformatted text preview: Lectures 1920: White Dwarfs and Neutron Stars My presentation is based on Chapters 2 and 3 of Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects by Stuart L. Shapiro and Saul A. Teukolsky. I. Relativistic Momentum and Energy (See 4.4 of Carroll & Ostlie.) Consider a particle with rest mass m moving with velocity v . The relativistic effects are characterized by = 1 p 1- v 2 /c 2 We can think of the moving particle as having an effective mass of m . Specifically, the momentum is p = mv The total energy (including the rest mass energy) is E = mc 2 = ( p 2 c 2 + m 2 c 4 ) 1 / 2 Notice: p E = v c 2 v = pc 2 E = pc 2 ( p 2 c 2 + m 2 c 4 ) 1 / 2 (1) II. Phase Space Distribution Function Recall that phase space is an abstract space in which each possible state of a system is represented as a unique point. For a system of particles, the important properties are position and momentum, so phase space has six dimensions ( x , y , z , p x , p y , and p z ). If we have a collection of particles, the way they are distributed in phase space is called the distribution function , F = number of particles per unit phase space volume We can use it to calculate various things. For example, if we want to know the total number of particles per unit volume in real space, without regard to their momenta, this is n = number of particles per unit volume of real space = Z F d 3 p The average of any quantity Q is h Q i = R Q F d 3 p R F d 3 p 1 For example, the mean square velocity is v 2 = R v 2 F d 3 p R F d 3 p = 1 n Z v 2 F d 3 p (2) The flux of any quantity Q is (draw picture): flux = amount flowing per unit area, per unit time = 1 Adt Z d 3 p F A v dt Q 1 3 = 1 3 Z Q v F d 3 p At a microscopic level, pressure is momentum flux, so P = 1 3 Z p v F d 3 p (3) This is known as the pressure integral (see 10.2 of Carroll & Ostlie). III. Ideal Non-degenerate Gas Lest this seem too abstract, lets consider how this formalism applies to the case of an ideal, non-degenerate, non-relativistic gas. In this case p = mv so the pressure integral is P = 1 3 m Z v 2 F d 3 p = 1 3 m v 2 n where I used eq. (2). We have already computed the mean square velocity for the Maxwell- Boltzmann distribution, h v 2 i = 3 kT/m . This then yields P = nkT We see how the distribution formulation leads to the same ideal gas law. IV. Cold, Ideal, Degenerate Gas For quantum particles, we think of phase space as discretized into little cells. In a problem with one spatial dimension, phase space is 2-d and each cell has area h . With three spatial dimensions, the 6-d volume of a cell in phase space is h 3 . In general (especially when the temperature is low), the particles will want to arrange them- selves to have the lowest total energy. Since energy is related to momentum, the particles will 2 want to have the lowest momentum possible. But there is a catch: the Pauli exclusion princi- ple limits the number of particles we can put in each phase space cell. For spin-1/2 particlesple limits the number of particles we can put in each phase space cell....
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lec-19-20 - Lectures 1920: White Dwarfs and Neutron Stars...

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