Quantum Statistical Mechanics for Stellar Astrophysics

QSM for Stellar AstrophysicsPHYS813: Quantum Statistical Mechanics

Quantum Statistical Mechanics for Stellar Astrophysics

Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A.

http://wiki.physics.udel.edu/phys813


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?

total gravitybutdE Q L E E Udt

= − = + stability of stellar equilibriumensured by gravitational force


Hertzsprung–Russell Diagram and Evolution of Solar-Mass Stars


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.


Internal Structure of Post-Main Sequence Stars in Pictures


Pressure due to Classical Thermal Motion Acts Against Gravity Collapse in Main Sequence Stars

2GMUR

≈ − star wants to contract to a state of lower energy (i.e., larger negative values of U), unless there is outward direct pressure to resist the contraction

BPV Nk T= in ordinary stars with fuel for thermnuclear fusion, outward pressure is provided bythermal motion

2B BNmk T k TGM

R Vm mρρ

≈ =

2

BGM U Nk T

R≈ − ≈

gravitation pressure at the center of the star is equal to thermal pressure

total thermal energy is comparable to the magnitude of total gravitational potential energy


White Dwarfs Stabilized by Fermi Pressure of Non-Relativistic Electrons

2 32 2

, , ,,

32 2 835

p e

eFF

e e

e e F

A Z A Z x Z AM Nm N xN

Np hm m V

E N

επ

ε

− =≈ =

= =

=

atomic number, number of protons, number of neutrons, electron fraction

mass of star, number of electrons

Fermi energy of electron gas in non-relativistic approximation

Total kinetic energy of electron gas

Total energy of cool star where thermal energy can be neglected 2 233( )

5 5p

e F

GN mE R N

Rε= −

2 32

2 2

9 1( ) 04 4e p

d xhE R R xNdR m GNMπ

= ⇒ =

White dwarfs cool off and contract to this radius

357 610.85 , 10 , 8000 km, =3 10 g cm2

M M N x R ρ= = = ⇒ ≈ ×


White Dwarfs Stabilized by Fermi Pressure of Relativistic Electrons

1 3

2 2 2 4 2

38

F e

F

p m c

F F e e F

Np hV

c p m c m c p c

π

ε

=

= + − →

energy per electron increases with N, so when it becomes comparable to

mc2 we have to switch to relativistic energy-momentum dispersion

1 3 2 2

2

33 9( )8 4 5

pGN mxNhcE R xNR Rπ

= −

for simplicity we assume uniform density assumed, while in reality density is larger in the

center of the star than further out

( ) ( )1 3 2 2

3 21 2 2 22

33 9 3( ) 0 125 28 4 5 16

c pcc c p

GN mxN hcE R xN N x hc GmR R

π ππ

= ⇔ = ⇒ =

both terms depend on R, so total anergy of star decreases continously with decreasing radius

570 2.2 10N = ⋅

2

00.7 1.40.5c c pxN N N m M = ⇒ =

numerically exact result called “Chandrasekhar limiting mass” as the largest mass a white dwarf can have and still cool off to a stable cold state

with finite radius and density


Neutron Stars Stabilized by Fermi Pressure of Neutrons

3 36 3 11 3 15 38 0.97 8 610 g cm 10 g cm 10 g cm

3 3(1 ) 1e n

p p

e p n

m c m cm mx h x x h x

ν

π πρ

− + → +

= × < < = × − −

E(R) could be lowered still further by changing through inverse beta decay which requires electrons with large kinetic energy

density of star at which Fermi momentum of electrons equals mec

neutrons become more abundant than protons at these densities

neutrons remain non-relativistic below this density

neutron electron gravitation

2 3 1 3 2 22

2 2 2

( , )

33 (1 ) 9 3 9(1 )5 8 4 8 4 5

p

p

E R x E E E

GN mx Nh xNhcx N xNR m R Rπ π

= + +

− = − + −

( )2 32 12

2

14 3

904 4( , ) 0; ( , ) 0

; 12.6 km; =2.4 10 g cm 0.5%

pp

hx R N GNmmE R x E R x

R xM M R x

π

ρ

− = ⇒ =∂ ∂ = = ⇒ ∂ ∂ = = × ⇒ =

neglects energy of protons (since they are small fraction of nucleons in neutron star) and nuclear forces

since it is composed of 1-x=99.5% of neutrons, such star is effectively a giant atomic nucleus

x

neutrons star radius is about 1000 times

smaller than the white dwarf radius


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


Quark Deconfinement in High-Density Matter Neutron Star

Mod. Phys. Lett. A

29, 1430022(2014)

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

22

2 1 103 3 2

n d u

u d u d

n d u

n d u

n n n n

P P P

µ µ µ→ += +

− = ⇒ =

= +

chemical equilibrium

charge conserved

pressure equilibrium

neutron density > 10 x 0.15 fm-3


Quark Matter Core in Massive Neutron Stars from Gravitational Waves Observed by LIGO + QCD Calculations


Insufficient Pressure → Black Holes and Their Thermodynamics

2

0

0c MGR

εε

− ≤

photon of energy ε cannot escape the surface of the star

to reach an observer c

0 2

2GMRc

=factor 2 comes from general relativity which must be used when

gravitation potential energy is comparable to other energies

Schwarzschild radius

3

4Bk cS AG

=

Bekenstein-Hawking entropy of a black hole

0 3 km for R M M≈ =

23

2 2 3

8

81 1 244

6.169 10 K

B Bk c k GS GM MT E c M G c c

MT

M

ππ

−

∂ ∂ = = = ∂ ∂

= ×

( )42 4 2 3

4 23 2 4

1 16 160 8

B

B

kE d G cT McA t dt c c k G M

π πσπ

∂= − ⇒ = − ∂

42 3 1/3 74

2 ( ) ( 3 ) 0 after 2.2 1015360

dM cM b M t M bt sdt G

τ= − ≡ − ⇒ = − → ≈ ×

Hawking radiation evaporates black hole

much longer than the age of the Universe ~1018 s

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

Quantum Statistical Mechanics for Stellar Astrophysics

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