Neutron Stars 1: Basics

Andreas ReiseneggerESO 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• Stuart L. Shapiro & Saul A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars, Wiley (1983): quite outdated on the phenomenology, but still the most comprehensive and pedagogical discussion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

Fpp ||

3hpppzyx zyx

mV

NnnppV

hNFF

,)3(3

4

23

1233

Each orbital has a phase-space volume

Ground state of system of N fermions of spin ½ (sz = ½) in a spatial volume V

(“box”):

Ground state (T=0) has all particles in the orbitals of the lowest possible energy

Fermi sphere in momentum space:

Degeneracy pressure• Fermi energy:

– Max. fermion energy @ T= 0

– Also chemical potential of the system: F = (T= 0)

– General:

– Non-relativistic limit, pF<<mc:

– Extreme relativistic limit, pF>>mc:

• Total energy E: Sum over Fermi sphere (all particles)

• “Degeneracy pressure” P: – Through kinetic theory or thermodynamics

– Non-zero value at T=0

– Non-relativistic Extreme relativistic

• Zero-temperature limit is good approximation as long as kT << F

– Thermal effects are only a small correction for white dwarfs & neutron stars

2222 )( cpmc FF 322

222 )3(

22n

mm

pmc F

F

312 )3( nccpFF

)()( PnPV

EP

N

m

nP

5

)3( 35322

34312 )3(4

1cnP

White dwarfsHydrostatic equilibrium

Non-relativistic electron

degeneracy pressure

Combining,

white dwarfs get smaller (denser & more relativistic) as M increases

Relativistic electrons

Combining with hydrost. equil. unique mass

(Chandrasekhar 1931)

4

2

2~

R

GMP

r

GM

dr

dP

35

3

3235322

~5

)3(

Rm

M

mm

nP

iee

343135

32

cmton1~km10~~ MmGm

Rie

34

334312 ~)3(

4

1

Rm

MccnP

i

Sun2

23

harChandrasek 4.1)(

~ Mm

GcM

i

Neutrons

1932: James Chadwick discovers the neutron.

Neutrons: decay or not decay?

• In vacuum (lab), neutrons decay with half-life ~ 15 min:

• In very dense matter, neutrons are stable (don’t decay) because low-energy proton & electron orbitals are already occupied

• “Chemical” (weak interaction) equilibrium

• 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.

eepn

epn

epn

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

CollapseCollapse of stellar core

huge density forces p + e n +

Neutrinos () escape: a few detected 2 hours before the light of SN 1987A.

Neutrinos

Remnant

BUT: No neutron star found!

Neutron stars• First approximation: Self-gravitating ball of non-

interacting neutrons at T=0

• Recall non-relativistic white dwarf:

• Neutron star (by analogy / scaling):

343135

32

cmton1~km10~~ MmGm

Rie

393138

32

cmton10~km10~~ MGm

Rn

Derived quantitiesAssume

• Surface gravity

• Escape speed

• “Breakup” rotation

• Schwarzschild radius

• “Relativity parameter”

Mass reduction ~ 20% when NS forms (carried away by neutrinos, perhaps gravitational waves)

• Gravitational redshift factor

km10,4.1 SunChandra RMMM

Earth11

2102 g

R

GMg

cRGMv 6.02esc

km42

2Schw c

GMR

4.0||2

c

22

grav2

escSchw2

Mc

Wv

R

R

Rc

GM

3.12

1121

2

Rc

GMz

ms)5.0(2kHz223Kepler RGM

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?

Mass-radius

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)

• 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

keV1.0~K105.14 41Sun

642 LLTTRL

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)

• Rotation of Crab pulsar

– 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

311

2

3

cm

g10

4

3

34

G

GRGM

Masses of pulsars in

binary systems

Fig. by I. Stairs,reproduced inLorimer 2005

“Tommy” GoldFranco Pacini

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...