Quantum Statistical Mechanics for Stellar Astrophysics
Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A. http://wiki.physics.udel.edu/phys813
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Light Emitted by Stars: Why is the Radiation from the Sun so Stable? STELLAR ASTROPHYSICS: four fundamental forces of nature come into play in a characteristic and spectacular manner → the stars are formed by a collapse of matter caused by gravitational attraction; the light that they emit is generated by electromagnetic interactions; strong interactions provide their main source of energy; and weak interactions contribute in a crucial way to make their lifetime so long. Sun as a main sequence star:
1937 Bethe and von Weiszäcker: nuclear fusion of hydrogen into helium that take place in the central part of the Sun produce some amount of heat per unit time, which is exactly equal to the luminosity because the state of the Sun is stationary. However, such reactions are very sensitive to small changes in the temperature: they are activated by a rise, hindered by a decrease. Thus, if it happens at some instant that a little more power is produced in the core than what is evacuated by radiation from the surface, why does the internal temperature not rise, eventually resulting in an explosion of the Sun? Conversely if the opposite perturbation occurs, why does the Sun not become extinct?
dE stability of stellar equilibrium =−=+QLbut E EU ensured by gravitational force dt total gravity PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Hertzsprung–Russell Diagram and Evolution of Solar-Mass Stars
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Nucleosynthesis and Fusion Reactions inside Main Sequence Stars
In both proton-proton chain and CNO cycle Coulomb repulsion must be overcome to initiate fusion, which involves quantum tunneling and requires extremely high kinetic energy of fusing particles.
The CNO cycle becomes the chief source of energy in stars of 1.5 solar masses or higher.
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Internal Structure of Post-Main Sequence Stars in Pictures
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Pressure due to Classical Thermal Motion Acts Against Gravity Collapse in Main Sequence Stars
2 GM star wants to contract to a state of lower energy (i.e., larger negative values of U), U ≈− unless there is outward direct pressure to resist the contraction R
in ordinary stars with fuel for thermnuclear fusion, outward PV= NkB T pressure is provided bythermal motion
ρGM NmkTρ kT ≈=BBgravitation pressure at the center of the star 2R Vm m is equal to thermal pressure
GM 2 ≈−U ≈ Nk T total thermal energy is comparable to the magnitude R B of total gravitational potential energy
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics White Dwarfs Stabilized by Fermi Pressure of Non-Relativistic Electrons AZ,, A−= Z , x Z A atomic number, number of protons, number of neutrons, electron fraction mass of star, number of electrons M≈= Nmpe, N xN
2 2 23 pF h 3Ne ε F = = Fermi energy of electron gas in non-relativistic approximation 2mmVee 28π 3 EN= ε Total kinetic energy of electron gas e5 eF
22 3 3GN mp ER()= Nε − Total energy of cool star where thermal energy can be neglected 55eF R
23 d xh2 91 =⇒= White dwarfs cool off and contract to this radius ERR() 0 22xN dR 44mepπ GNM
1 3 M=0.85 MN , = 1057 , x =⇒≈ R8000 km, ρ=3 × 106 g cm 2
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics White Dwarfs Stabilized by Fermi Pressure of Relativistic Electrons
13 3N phF = 8πV energy per electron increases with N, so when it becomes comparable to pFe mc ε =2 2 +− 24 2 → mc2 we have to switch to relativistic Fc p Fe mc mc e p F c energy-momentum dispersion
13 22 for simplicity we assume uniform density 39xNhc 3GN mp = − assumed, while in reality density is larger in the ER() 2 xN 84RRπ 5 center of the star than further out both terms depend on R, so total anergy of star decreases continously with decreasing radius
13 22 39xN hc 3GN m 312 32 =⇔ c =cp ⇒= ππ22 ER( ) 0 2 xNccN (125) x( hc 2 Gmp) 8RR 4π 5 16 57 N0 =2.2 ⋅ 10
2 numerically exact result called “Chandrasekhar x limiting mass” as the largest mass a white dwarf Nc=0.7N0 ⇒= Nmcp 1.4 M can have and still cool off to a stable cold state 0.5 with finite radius and density
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Neutron Stars Stabilized by Fermi Pressure of Neutrons epn− + →+ν E(R) could be lowered still further by changing x through inverse beta decay which requires electrons with large kinetic energy 33 8ππmcen0.97 6 3 11 3 8 mc 6 15 3 mmpp= ×10 g cm 10 g cm <<ρ = ×10 g cm 3xh x 3(1−−x ) h 1 x density of star at which Fermi neutrons remain non-relativistic below this density momentum of electrons equals mec neutrons become more abundant than protons at these densities ERx(,)=++ E E E neglects energy of protons (since they are small fraction neutron electron gravitation of nucleons in neutron star) and nuclear forces 23 13 3 (1− x ) Nh2 9 3xNhc 9 3GN22 m = −+ −p 22(1x ) N 2xN 58Rmp 4ππ 8R 4 5R
2 23 neutrons star radius h 9 −1 =⇒= 2 is about 1000 times ∂∂ x0 R 2 N( GNmp ) smaller than the ERx(,)= 0; ERx (,)= 0 ⇒ 44mp π ∂∂Rx white dwarf radius 14 3 M= MR ; = 12.6 km; ρ=2.4× 10 g cm ⇒=x 0.5% since it is composed of 1-x=99.5% of neutrons, such star is effectively a giant atomic nucleus PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Observing Pulsar-Type Neutron Stars Through Their Radiation
NASA Chandra satelite imaging of rings created by the X-rays from a Circinus-X1 double star system, containing neutron star orbiting around another massive star, reflecting off different dust clouds
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Quark Deconfinement in High-Density Matter Neutron Star
n→+2 duneutron density > 10 x 0.15 fm-3
µn=2 µµ du + chemical equilibrium
21 1 Mod. Phys.Lett. A nn− =⇒=0 n ncharge conserved 33ud u 2 d pressure equilibrium PPPn= du + At the density and pressure at which a neutron–quark phase transition is expected to occur, the system of neutrons is not ideal degenerate Fermi gas 29 , 1430022 , (2014)
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Quark Matter Core in Massive Neutron Stars from Gravitational Waves Observed by LIGO + QCD Calculations
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics Insufficient Pressure → Black Holes and Their Thermodynamics
2 photon of energy ε cannot ≈= ε cM escape the surface of the star R0 3 km for MM ε −≤G 0 to reach an observer c 2GM R0 = Schwarzschild R0 2 radius factor 2 comes from general relativity which must be used when c gravitation potential energy is comparable to other energies 2 11∂∂S kc3 2GM 8π kG 3 = = BBπ = 24 23M kcB Bekenstein-Hawking T∂∂ EcM4 G c c SA= entropy of a black hole 4G M T = 6.169 × 10−8 K M
HAWKING RADIATION: Particle-antiparticle pairs created from vacuum normally quickly annihilate each other, but near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation 4 1 ∂E d π 24k 16πGc231 =−⇒σ 42 =− B T( Mc ) 32 4 Hawking radiation evaporates black hole A∂ t dt 60 c c8π kB G M
4 2 dM c 3 1/3 74 M =− ≡−b ⇒ M( t ) = ( M − 3 bt ) → 0 after τ ≈2.2 × 10 s much longer than the age of dt15360 G2 the Universe ~1018 s
PHYS813: Quantum Statistical Mechanics QSM for Stellar Astrophysics