Neutron Stars
1: Basics
Andreas Reisenegger
ESO Visiting Scientist
Associate Professor,
Pontificia Universidad Católica
de Chile
Outline of the Lecture Series
1. Basics: Theory & history: prediction, discovery, etc.
2. Phenomenology: The many observational
“incarnations” of NSs & what we can learn from
them
3. Thermal evolution: Cooling & heating
mechanisms, expected thermal history, obs.
constraints & what they tell us about nuclear physics
& gravity
4. Magnetism: Determination of NS magnetic fields,
their origin, evolution, and related physical processes
Outline of Lecture 1
• Degenerate fermions, white dwarfs, &
Chandrasekhar mass
• Prediction of neutron stars, main predicted
properties
• Pulsar discovery & interpretation
Bibliography - 1
• Stuart L. Shapiro & Saul A. Teukolsky, Black Holes,
White Dwarfs, and Neutron Stars, Wiley (1983): quite
outdated on the phenomenology, but still the most
comprehensive and pedagogical discussion
• Richard R. Silbar & Sanjay Reddy, Neutron Stars for
Undergraduates, Am. J. Phys. 72, 892-902 (2004;
erratum 73, 286, 2005), nucl-th/0309041: how to build
simple numerical models of neutron stars
• James M. Lattimer & Madappa Prakash, The Physics
of Neutron Stars, Science, 304, 536 (2004): review
• Norman K. Glendenning, Compact Stars: Nuclear
Physics, Particle Physics, and General Relativity,
Springer (1997): quite theoretical
Bibliography - 2
• Kip S. Thorne, Black Holes & Time Warps: Einstein’s
outrageous legacy, Norton (1993): entertaining popular
history of the idea of compact stars & black holes
• Bernard F. Schutz, A first course in general relativity,
Cambridge (1985): rigorous, but elementary account of
GR from the basics up to relativistic stars, black holes, &
cosmology
• Soon to appear: P. Haensel, A. Y. Potekhin, & D. G.
Yakovlev, Neutron Stars 1: Equation of State and
Structure (2006)
Pauli
principle
Fermions:
• particles of half-integer
spin (½, 3/2, ...):
– electrons, protons,
neutrons...
• obey Pauli exclusion
principle (1925): No more
than 1 fermion can occupy
a given orbital (1-particle
quantum state)
 “Fermi-Dirac statistics”
Ground state of fermion
system
Ground state (T=0) has all particles in the
orbitals of the lowest possible energy
 Fermi sphere in momentum space:

| p |  pF
Each orbital has a phase-space volume
x y z px p y pz  h3
Ground state of system of N fermions of spin ½ (sz =  ½) in a spatial volume V
(“box”):
N  h3
4
 V   pF3
2
3

pF  (3 2 n) 3 , n 
1
N 

V m
Degeneracy pressure
• Fermi energy:
–
–
–
–
Max. fermion energy @ T= 0
Also chemical potential of the system: F =  (T= 0)
General:  F  (mc 2 ) 2  pF2 c 2
2
2
Non-relativistic limit, pF<<mc:   mc2  pF   (3 2 n) 2 3
F
2m 2m
 F  pF c  c(3 2 n)1 3
• Total energy E: Sum over Fermi sphere (all particles)
• “Degeneracy pressure” P:
– Extreme relativistic limit, pF>>mc:
 E 
P  
  P ( n)  P (  )
– Through kinetic theory or thermodynamics
 V  N
– Non-zero value at T=0
1
2
23 53
2 13
43
– Non-relativistic P  (3  ) n
Extreme relativistic P  (3 ) cn
4
5m
• Zero-temperature limit is good approximation as long as kT << F
– Thermal effects are only a small correction for white dwarfs & neutron stars
White dwarfs
dP GM

 2
dr
r
Hydrostatic equilibrium
Non-relativistic electron
degeneracy pressure
Combining,
GM 2
 P~
R4
(3 2 ) 2 3 n 5 3  2 3  M 


P
~
5me
me  mi R 3 
53
2 3
1 3
4
3
R~
M
~
10
km


~
1
ton
cm
Gme mi5 3
 white dwarfs get smaller (denser & more relativistic) as M increases
Relativistic electrons
 M 
1

P  (3 2 )1 3 cn 4 3 ~ c
3 
4
 mi R 
43
Combining with hydrost. equil.  unique mass M
Chandrasekhar
(Chandrasekhar 1931)
(c G )3 2
~
 1.4M Sun
2
mi
Neutrons
1932: James Chadwick
discovers the neutron.
Neutrons:
decay or not decay?
• In vacuum (lab), neutrons decay with half-life ~ 15 min:
n  p  e  e
• In very dense matter, neutrons are stable (don’t decay) because low-energy
proton & electron orbitals are already occupied
• “Chemical” (weak interaction) equilibrium
n pe
 n   p  e
• Around nuclear density, neutrons coexist stably with a much smaller
number (~1%) of protons & electrons (fraction density-dependent &
uncertain)
• At higher densities, strong interactions among particles are difficult to
model, making the state of matter (& eq. of state) more & more uncertain.
Baade & Zwicky (1934):
“With all reserve we advance the view that supernovae represent
the transition from ordinary stars into neutron stars, which in their
final stage consist of extremely closely packed neutrons.”
Supernova 1987A (23 Febr. 1987) in the Large Magellanic Cloud:
before & after
Collapse
Collapse of stellar core
 huge density
forces p + e  n + 
Neutrinos () escape: a
few detected 2 hours
before the light of SN
1987A.
Neutrinos
BUT: No neutron star found!
Remnant
Neutron stars
• First approximation: Self-gravitating ball of noninteracting neutrons at T=0
• Recall non-relativistic white dwarf:
2 3
1 3
4
3
R~
M
~
10
km


~
1
ton
cm
Gme mi5 3
• Neutron star (by analogy / scaling):
2 3
1 3
9
3
R~
M
~
10
km


~
10
ton
cm
Gmn8 3
Derived quantities
Assume
M  M Chandra  1.4M Sun , R  10 km
• Surface gravity
g
• Escape speed
GM
11

2

10
g Earth
2
R
vesc  2GM R  0.6 c
• “Breakup” rotation
• Schwarzschild radius
Kepler  GM R3  2  2 kHz  2 (0.5 ms)
RSchw 
• “Relativity parameter”
2GM
 4 km
c2
2
2 | Wgrav |
2GM RSchw  vesc 



 0.4


c2R
R
Mc 2
 c 
 Mass reduction ~ 20% when NS forms (carried away by
neutrinos, perhaps gravitational waves)
• Gravitational redshift factor
 2GM 
1  z  1  2 
c R 

1 2
 1.3
Relativistic stellar structure eqs.
Since P=P() only (no dependence on T, equilibrium composition),
these two equations are enough to calculate the NS or WD structure.
dL/dr=..., dT/dr=... are important only for the thermal evolution.
First (numerical) solution for NSs: Oppenheimer & Volkoff 1939
The state of matter
changes with
density (T  0):
• “Ordinary” solid
• Solid + neutrons
• npe liquid
• More exotic
particles:
–
–
–
–
muons 
mesons , 
hyperons , 
???
• Quark matter?
Massradius
relation
for NSs
Black (green) curves are for normal matter (SQM) equations of state.
Regions excluded by general relativity (GR), causality, and rotation constraints are indicated.
Contours of radiation radii R are given by the orange curves.
The dashed line labeled I/I = 0.014 is a radius limit estimated from Vela pulsar glitches (27).
from Lattimer & Prakash 2004
Expected thermal radiation
• Neutron stars are born in stellar core collapse: initially very hot, expected to
radiate thermally (~ blackbody) (Chiu 1964, Chiu & Salpeter 1964)
L  4 R  T
2
4
 T  1.5 10 L LSun  K ~ 0.1keV
6
14
• Might be detectable in soft X-rays (difficult)
• Start of X-ray astronomy (Giacconi et al. 1962):
– several sources (quasars, etc.)
– growing theoretical interest in NSs
– at first no unambiguous detections of NSs (X-ray pulsars found by UHURU, 1971)
– no detections of pure, thermal radiators until much later
Pulsars: discovery
1967: PhD student Jocelyn Bell &
her supervisor Anthony Hewish
detect a very regular Pulsating
Source of Radio (PSR 1919+21)
with P=1.377 s, initially “LGM 1”.
http://www.jb.man.ac.uk/~pulsar/Education/Sounds/sounds.html
Pulsar in the Crab nebula
(remnant of SN 1054)
Lyne 2000
How we know pulsars are NSs
• Association with supernova remnants (SNRs)
3
• Rotation of Crab pulsar   GM R  4 G 3

3 2
11 g

 10
4 G
cm 3
– Much faster ones (“millisecond pulsars”) found later
• Energy budget of SNR in rough agreement with energy lost
from rotating NS.
• Very high-energy (non-thermal) emission: likely relativistic
system
• Thermal emission (X-rays): emitting region  10 km
• Binary systems: Precise masses 1.25-1.45 Msun ~ MChandra
Masses of
pulsars in
binary
systems
Fig. by I. Stairs,
reproduced in
Lorimer 2005
Franco Pacini
“Tommy” Gold
Known pulsars: ~ 2000
Other neutron stars
Several other classes of observed objects are also believed to
be NSs:
• X-ray binaries (some NSs, some black holes)
• “Soft gamma-ray repeaters” & “Anomalous X-ray Pulsars”
 “Magnetars”
• Thermal X-ray emitters:
– bright central objects in SNRs
– fainter, isolated objects
• Radio transients (RRATs)
See tomorrow’s Lecture for details...