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Home , Stellar classification, Stellar structure

Dwarfs - Degenerate Stellar Configurations

Austen Groener Department of - Drexel University, Philadelphia, Pennsylvania 19104, USA II

May 17, 2010

Abstract The end product of low-medium mass is the degenerate stellar configuration called a . 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 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 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 exerted by each (this is in the due to a large birthrate of their pro- where (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 . 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 : Pressure Gradient = (classification which aligns 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 more dense than the electron degenerate gas. distribution of mass.

II Stellar Pressure and Hydrostatic Equi- III Ion Pressure librium We start with the for an ideal ion In the field of , 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 and , 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 , 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 . 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 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 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 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: ∞ 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 hν 1 4 gen and 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 , 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 ) 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 leaving behind a 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 is halted by roughly 108K the core begins to fuse helium into the electron degeneracy pressure (Fig.1), an applica- carbon and through what is known as the tion of the Pauli Exclusion Principle. Since electrons triple-. 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 is higher by more than electrons. Complete degeneracy occurs when elec- an order of 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 ). 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. 2 h 3 2 1 ρ 5 3 3 3 3 Pe−,deg = ( ) 5 ( ) (22) ∆V ∆ p ≥ h (28) 20me− π (mH ) 3 µe−

Where p0 is the maximal momentum: Eventually as the density becomes sufficiently high

3 the range of momenta exceed the range of momenta 3h ne 1 p0 = ( ) 3 (23) corresponding to the gas temperature. Once this oc- 8π curs, equation (28) is minimized to a strict equality. This marks the level at which quantum mechanics must be considered to be of equal importance to clas- sical thermodynamics. White dwarfs differ from main-sequence stars in another very important way. They are considered to be highly under-luminous. Judged with reference to other stars of comparable mass, they exhibit very low average . This clearly has mostly to do with a lack of nucleosynthesis, although it may be partly due to another characteristic of the degener- ate gas it is composed of. A degenerate electron gas behaves much like a metal, conducting heat very ef- ficiently. One known consequence of this will be a nearly uniform internal temperature - the tempera- ture gradient is very small.

Figure 2: An Infinite Square Well of Length L dT 3 κρ F = − (29) dr 4ac T 3 4πr2

Let us now consider visually three electrons in a This is the radiative transfer equation, where a is F 1-D square well (Fig. 2). Shown here are the wave- the radiation constant, F ≡ Luminosity, 4πr2 ≡ functions of these electrons in the ground state. F lux, and: a b κ = κ0ρ T (30) h mev = p = (24) λ VIII Further Degeneracy 2 2 2 p1 p2 p3 E = + + (25) Sufficiently dense matter containing protons will 2me 2me 2me also experience proton degeneracy pressure. Pro- h2 1 1 1 3h2 tons confined to a small enough region will have E = ( 2 + 2 + 2 ) = 2 (26) 2me (2L) (2L) L 4meL large uncertainties in their momenta. Due to a much larger mass than the electron, proton velocities will As the length of our box shrinks in size, the energy be smaller, representing only a small correction to of the particles will increase: the equations of state of an electron degenerate gas. dE 3h2 The equations governing electron degeneracy P ∝ | | = 3 (27) show us that a maximum mass for a non-rotating dL 2meL

4 white dwarf exists. This mass is the Chandrasekhar IX Conclusion limit - approximately 1.4M . Although seemingly in the last phase of their lives, carbon-oxygen white In this paper we have discussed hydrostatic equilib- dwarfs are capable of further fusion reactions when rium by piecing together the various components of part of a binary system with a . If the white pressure of a main-sequence star. We then devel- dwarf accretes mass from its companion, its core oped the sequence of processes through which low- temperature will reach the ignition temperature re- medium mass stars transition into quired for carbon fusion. The sudden initiation of nu- by making use of the Pauli Exclusion Principle and clear fusion causes a runaway reaction causing what the Heisenberg Uncertainty Principle. We have also is known as a supernova (Type Ia). Once exceeding made the distinction as to where quantum mechan- the mass limit, overcomes the electron degen- ics cannot be ignored in stellar calculations. Lastly, eracy pressure and begins to collapse. Depending on a qualitative description has been provided regard- how much mass the white dwarf accreted in the pro- ing the transition to other degenerate configurations cess, the end product is a neutron star or a . arising from white dwarf and neutron star binary ac- Not all neutron stars remain in isolation. In dense cretion. stellar regions such as globular clusters, neutron stars are able to capture a companion. Through a mass ac- X References cretion process (as with the white dwarf binary) the neutron star can transition to a black hole. 1. S. Chandrasekhar, An Introduction to the Study In either case, further degeneracy pressures be- of , Dover, New York, 1939. come more important than that of the electron. In the case of a neutron star, gravitational pressure is 2. D. Prialnik, An Introduction to the Theory counteracted by neutron degeneracy pressure. As the of Stellar Structure and Evolution, Cambridge density increases the Fermi Energy of the electrons University Press, Cambridge 2000. will increase to a point where it is energetically fa- 3. Luo Xin-Lian, Bai Hua, Zhao Lei, “Double vorable for them to combine with protons to produce Degenerate Stars ”, arXiv:0908.2276v1 [astro- neutrons. This occurs through an inverse beta-decay ph.SR] process. Similarly, for neutron stars there exists some up- 4. Christian Y. Cardall, “Degeneracy, the per limit to the mass as well, called the Tolman- virial theorem, and stellar collapse ”, Oppenheimer-Volkoff limit. Any more massive than arXiv:0812.0114v1 [astro-ph] this and we now move into the realm of quark stars (composed of, you guessed it, degenerate quark gas - if you can even call it that), and finally into the realm of black holes. Solving for this final limit requires the use of General Relativity, and is most commonly displayed as the Schwarzschild radius:

2GM(r) r = (31) s c2

The Schwarzschild radius is a characteristic radius which is associated with every massive body. It de- scribes the volume at which the object must be fully contained in such that there exists no known degen- eracy pressure to counteract its self-gravity. Essen- tially it describes a singularity: a point mass of in- finitely large density.

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