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White Dwarfs - Degenerate Stellar Configurations White Dwarfs - Degenerate Stellar Configurations Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics II May 17, 2010 Abstract The end product of low-medium mass stars is the degenerate stellar configuration called a white dwarf. Here we discuss the transition into this state thermodynamically as well as developing some intuition regarding the role of quantum mechanics in this process. I Introduction and Stellar Classification present at such high temperatures are small com- pared with the kinetic (thermal) energy of the parti- It is widely believed that the end stage of the low or cles. This assumption implies a mixture of free non- intermediate mass star is an extremely dense, highly interacting particles. One may also note that the pres- underluminous object called a white dwarf. Obser- sure of a mixture of different species of particles will vationally, these white dwarfs are abundant (∼ 6%) be the sum of the pressures exerted by each (this is in the Milky Way due to a large birthrate of their pro- where photon pressure (PRad) will come into play). genitor stars coupled with a very slow rate of cooling. Following this logic we can express the total stellar Roughly 97% of all stars will meet this fate. Given pressure as: no additional mass, the white dwarf will evolve into a cold black dwarf. PT ot = PGas + PRad = PIon + Pe− + PRad (1) In this paper I hope to introduce a qualitative de- scription of the transition from a main-sequence star At Hydrostatic Equilibrium: Pressure Gradient = (classification which aligns hydrogen fusing stars Gravitational Pressure. to a line on the Hertzsprung-Russell Diagram) to a dP GM white dwarf through the help of quantum mechanics. = − ρ (2) dR R2 In addition, I will also discuss possible mechanisms for the formation of other configurations of exotic This is of course assuming a spherically symmetric matter more dense than the electron degenerate gas. distribution of mass. II Stellar Pressure and Hydrostatic Equi- III Ion Pressure librium We start with the equation of state for an ideal ion In the field of Stellar Dynamics, we often define the gas: relation between the pressure exerted by a system PIon = nIonkT (3) of particles of known composition and its ambient Where nIon is the number of ions per unit volume temperature and density, P=P(ρ,T,X), to be called V. To obtain the total number of ions in this unit vol- the equation of state. Assuming stellar gas to be an ume, we must sum the following relation over all ion ideal gas, an assumption seemingly made on care- species: less grounds, will prove to be acceptable solely due ρ X to the fact that Coulomb interactions of ionized gases n = (4) mH A 1 X X ρ Xi n = n = (5) To clarify, our assumptions have been non- Ion i m A i i H i interacting gases and complete ionization. So far, we have still failed to mention anything regarding the Where mH is the atomic mass unit (by convention - conditions of stellar interiors such that these effects 1 of a carbon nucleus 6= proton mass or a hydrogen 12 cannot always be neglected. At a certain point, quan- atom), X is the mass fraction of a species, and A is tum mechanical and relativistic effects will enter the the baryon number of a species. The mean atomic picture. mass of stellar material µIon and the ideal gas con- stant are then defined respectively by: V Radiation Pressure 1 X Xi ≡ (6) µIon Ai The idea of a pressure is generally associated with a i F orce Area . This force can then be thought of as the sum k of massive particles imparting momenta upon the R ≡ (7) mH walls of some container. Although a photon does not have mass, its momentum is hν. Radiation pressure Finally we get: is due to the transfer of momentum to gas particles R whenever absorption and scattering of photons oc- PIon = ρT (8) µIon curs within the gas. In Thermodynamic equilibrium the photon distribution is isotropic and the number of IV Electron Pressure photons with frequencies in the range (ν; ν + dν) is given by the blackbody distribution: Starting in a similar fashion, the equation of state for 8πν2 dν n(ν)dν = (14) an ideal electron gas is: 3 hν c exp kT − 1 We can use the pressure integral to obtain an expres- Pe− = ne− kT (9) sion for the pressure due to radiation: 1 Where n − is the number of free electrons per unit 1 Z e P = νpn(p)dp (15) volume V. Here we concentrate our focus on the stel- 3 0 lar interior, where at temperatures of 106K, hydro- Z 1 1 hν 1 4 gen and helium are completely ionized. The total PRad = c n(ν)dν = aT (16) 3 0 c 3 number of electrons per unit volume is: Where a is the radiation constant: 8π5k4 4σ X ρ X Zi a = = (17) ne− = Zini = Xi (10) 15c3h3 c mH Ai i i As stated previously, each collision between a Where Z is the charge on the nucleus. We then define photon and an atom will excite the atom energeti- the average number of free electrons per nucleon, cally, transferring momentum in the direction of the −1 incoming photon. Inevitably returning to its original µe− , as: 1 X Zi state by photon emission, it recoils in the direction = Xi (11) µ − A opposite the initial photon. The direction of these e i i emitted photons are random and after a long series Which after some substitution will give us: of these interactions the random changes in momenta ρ due to emission cancel out and the net change in the ne− = (12) atom’s momentum is in the direction of the photon. µe− mH Putting all pressures together((7),(12) and(15)), The electron pressure can thus be written as: we achieve a total pressure of: 1 1 1 R P = ( + )RρT + aT 4 (18) Pe− = ρT (13) µIon µe− 3 µe− 2 VI Evolution of a Low-Medium Mass Star Throughout a star’s life on the main sequence, the star constantly battles to maintain stability. The rate at which stars fuse elements in their cores is very sen- ( −E ) sitive to temperature (e kT ). If the star were to be compressed even a little, the core would rise in tem- perature and the rate of fusion would increase con- siderably. This increased rate of fusion results in a higher production of photons (which can take on the order of 105 years to reach the photosphere) and con- sequently a higher photon pressure. Given enough fuel, this self-sustaining process will continue for the vast majority of the star’s life. Considering the topic at hand we focus our atten- tion to those stars we believe will end up as white dwarfs: 0.8M -8M . Stars with insufficient mass Figure 1: Hydrostatic Equilibrium will never reach a core temperature and density high enough to begin the fusion of hydrogen. Stars in ex- cess of roughly 8M will eventually produce cores Since the core is now completely devoid of an energy which exceed the Chandrasekhar limit and will result source, it begins its final descent into the white dwarf in a supernova leaving behind a neutron star or black stage. hole remnant. Once the hydrogen in the core has been exhausted, the star undergoes contraction due to lack of energy VII Electron Degeneracy Pressure production in the core. It then shrinks in radius, heat- ing up the core to higher and higher temperatures. At The final collapse of the stellar core is halted by roughly 108K the core begins to fuse helium into the electron degeneracy pressure (Fig.1), an applica- carbon and oxygen through what is known as the tion of the Pauli Exclusion Principle. Since electrons triple-alpha process. This new production of energy are fermions, no two can occupy identical quantum will temporarily re-stabilize the star. This phase of states. the electrons are forced into higher and higher stable helium burning is significantly shorter than the energy levels causing them to be relativistic. Even at main-sequence phase of hydrogen burning. This is zero temperature, there exists a non-zero energy due essentially due to two reasons: to these relativistic electrons. This is referred to as • The fusion of helium into carbon and oxygen the Fermi Energy. provides roughly 1 the energy per unit mass 10 1 1 supplied by hydrogen burning. Since an election can have two spin states (+ 2 ; − 2 ) each location in phase space can have at most two • The stellar luminosity is higher by more than electrons. Complete degeneracy occurs when elec- an order of magnitude compared with the main- trons are forced to occupy all available momentum sequence luminosity of the same star. states. Applying the Heisenberg and Pauli Princi- ples to a completely degenerate isotropic electron gas Inevitably, the Helium fuel source will run out as yields the momentum distribution (number of elec- well, causing another phase of contraction as well trons with momenta in the interval (p; p + dp) per as an increase in temperature in the core. This time unit volume): the remaining hydrogen in the shell outside of the core will proceed to burn, separating the core from 2 n − (p)dp = (19) its remaining envelope (this is the planetary nebula). e ∆V 3 2 2 In this case we are doing work against degeneracy ne− (p)dp = 4πp dp (20) h3 pressure. After a similar process, we achieve an electron de- generacy pressure of: According to the Heisenberg Uncertainty Prin- ciple as we restrict the location of these electrons 8π 5 Pe−;deg = 3 p0 (21) by decreasing the size of the box, the uncertainty in 15m − h e their momentum will increase.
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