Thomas Tauris  AIfA Bonn Uni. / MPIfR
Bonn, Summer 2014

1:  Introduction
 Degenerate Fermi Gases
Non-relativistic and extreme relativistic electron / (n,p,e-) gases

2:  White Dwarfs
Structure, cooling models, observations

3:  Neutron Stars
Structure and Equation-of-state
 Radio Pulsars
Characteristics, observations, spin evolution, magnetars

4:  Binary Evolution and Interactions
Accretion, X-ray Binaries, formation of millisecond pulsars
 Black Holes
Observations, characteristics and spins

5:  Testing Theories of Gravity Using Pulsars
 Gravitational Waves
Sources and detection
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
2

Structure of WDs





EoS below neutron drip







Basic characteristics
Stability
Super-Chandrasekhar mass WDs
Chandrasekhar mass limit
Neutron-rich nuclei
Neutron drip
Semi-empirical mass formula
Including shell effects and lattice energy
Harrison-Wheeler EoS
Baym-Pethick-Sutherland (BPS) EoS
Observations
Bonn, Summer 2014

Surface layers

Photon diffusion equation L  4 r 2






Temperature gradient
Pressure gradient (via hydrostatic equilibrium)
Core-surface boundary conditions
Luminosity as a function of (M, T)
Residual ion thermal energy
Cooling age
Crystallization



d
aT 4
3 dr
Elementary treatment of WD cooling



c
Rapid cooling
Observational support of WD cooling models
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
4
Structure of WDs
Surface layers: (H) He
non-degenerate layers in ”radiative equilibrium”
LTE with outward energy flux by diffusion of photons
(only small gradient in net flux, I: Planck function)
Interior: CO, ONeMg (He WDs in close binaries)
electrons are completely degenerate
 electrons have a large mean free path, e
because of the filled Fermi sea
 thermal conductivity, th is large
 temperature, T is uniform (isothermal core)
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
5

Photon diffusion equation:




L  4 r 2
c

d
aT 4
3 dr

Temperature gradient
Pressure gradient (via hydrostatic equilibrium)
Core-surface boundary conditions
Luminosity as a function of (M, T)

c


d
dT
aT 4

 ....
3 dr
dr
dP
m( r )
  g   G 2 
hydrostatic equil.
dr
r
L   4 r 2
b-f photoionization of atoms
f-f inverse bremstrahlung of e-
dP
 ....
dT

Integrate…

   0 T 3.5

P
M 17/4
(1) P 
T
L
Bonn, Summer 2014

 mu
Kramer's opacity

 x/
,
 I ( x)  I 0 e


1 
 
mean free path
kT
ideal gas

Boundary conditions:
P  0, T  0, m(r )  M (thin envelope)
Thomas Tauris - Bonn Uni. / MPIfR
6
Core boundary condition (transition from surface layers to core region)
Pgas  Pdeg
Pgas 
M 17/4
T  K 5/3
L
(2) L  M T3.5
Typically:

kT =K5/3  5/3  T5/2
 mu

( L  C  M  T3.5 )
T  106  107 K

L  105  102 L
core temperature
Note
  103 gcm3
c

rsurface layers
Bonn, Summer 2014
RWD
Thus the assumption of a fully degenerate star (cold EoS) is valid!
(M,R)-relations obtained earlier are ok!
Thomas Tauris - Bonn Uni. / MPIfR
7
Loss of residual thermal energy of ions  radiation
thermal
Eion
  radiation
(
 Eethermal
 gas
)
cannot be deliberated because of the filled Fermi-sea
 neutrino emission is only important very early when T  108 K
3
2
(erg/K) for a monatomic gas (e.g. C -ions)
Specific heat capacity per ion: cv  k
Total thermal energy: U  cv  T  Nions
L
dU
dt
Integrate…
(T0
T)

C  M  T 3.5 

M 
  
 L

3
M
kT
2
Amu
  10
48
erg , T  107 K
d 3
M 
kT


dt  2
Amu 
5/7
cooling age!
Problem: estimated WD cooling ages were too large by a factor 10 WD
Bonn, Summer 2014

Thomas Tauris - Bonn Uni. / MPIfR
 cluster
8
Crystallization of ion lattice
Formation of lattice:
 (ri )2
ri
2
1

16

Ecoulomb Z 2 e2 / ri

 171
Ethermal
kT
Lindemann’s empirical rule
cv
lattice
3k
T  cv (lattice vibrations) dominate over cv (free thermal motion)
cv = 3k (including Epot,lattice  ½ k per mode)
ideal gas
3/2 k
Quantum-mechanical
effects
D
Bonn, Summer 2014
Tm
T  D
Tg
 cv    
T
Thomas Tauris - Bonn Uni. / MPIfR
rapid cooling
observational consequences
9
diamond WD!
”Lucy”

 Beatles song: ”Lucy in the Sky with Diamonds”
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
10
Althaus et al. (2007), A&A 465, 249
Massive 1.06-1.28 Msun ONeMg WDs with
helium/hydrogen envelopes
Hansen et al. (2007), ApJ 671, 380
The WD cooling sequence of NGC 639
Gilles Fontaine (2000)
Evidence for rapid cooling…

Surface layers

Photon diffusion equation L  4 r 2






Temperature gradient
Pressure gradient (via hydrostatic equilibrium)
Core-surface boundary conditions
Luminosity as a function of (M, T)
Residual ion thermal energy
Cooling age
Crystallization



d
aT 4
3 dr
Elementary treatment of WD cooling



c
Rapid cooling
Observational support of WD cooling models
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
13

1:  Introduction
 Degenerate Fermi Gases
Non-relativistic and extreme relativistic electron / (n,p,e-) gases

2:  White Dwarfs
Structure, cooling models, observations

3:  Neutron Stars
Structure and Equation-of-state
 Radio Pulsars
Characteristics, observations, spin evolution, magnetars

4:  Binary Evolution and Interactions
Accretion, X-ray Binaries, formation of millisecond pulsars
 Black Holes
Observations, characteristics and spins

5:  Testing Theories of Gravity Using Pulsars
 Gravitational Waves
Sources and detection
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
14
Shapiro & Teukolsky (1983), Wiley-Interscience
Curriculum
- Chapter 4: p.82-87, (91-92), 100-105
Exercises: #23, 24
- Wed. May 14 15:00-16:30, AIfA raum 0.006 !!
Bonn, Summer 2014
Thomas Tauris - Bonn Uni. / MPIfR
15