NEUTRON . but, at high densities, degenerate neutrons are ener- getic enough to produce new particles (hyperons and 1934: Baade and Zwicky proposed that supernovae repre- pions) → the pressure is reduced because the new par- sented the transition of normal stars to neutron stars ticles only produce a small pressure 1939: Oppenheimer and Volkoff published the first • theoretical model Effect of gravity (gravitational binding energy of neu- tron is comparable with rest ): 1967: discovery of by Bell and Hewish • Gravity is strengthened at very high densities and Maximum mass of a : pressures. Consider the pressure gradient:

• dP Gm Neutrons are fermions: degenerate neutrons are un- Newton: = − able to support a neutron star with a mass above a dr r2 certain value (c.f. Chandrasekhar mass limit for white 2 3 2 ¡

dP Gm (1 + P/( c ))(1 + 4 r P/(mc )) dwarfs). Einstein: = − × dr r2 1 − 2Gm/(rc2) • Important differences from white dwarf case: • Pressure P occurs on RHS. Increase of pressure, (i) interactions between neutrons are very important needed to oppose gravitational collapse, leads to at high densities strengthening of gravitational field ii) very strong gravitational fields (i.e. use General •

need an equation of state P( ) that takes account of Relativity) n − n interactions N.B. There is a maximum mass but the calculation • Oppenheimer and Volkoff (1939) calculated M for is very difficult; the result is very important for max a star composed of non-interacting neutrons. black-hole searches in our Galaxy Result: Mmax = 0.7 M¯ • Crude estimate– ignore interactions and General Rel- • Enhanced gravity leads to collapse at finite density ativity – apply same theory as for white dwarfs: when neutrons are just becoming relativistic – not ul- Mmax ' 6 M¯ trarelativistic • interactions between neutrons increase the theoretical • Various calculations, using different compressibilities maximum mass for neutron star matter, predict ∼ . The interaction is attractive at distances 1.4 fm Mmax ∈ [1.5, 3] M¯ RNS ' 7 − 15 km but repulsive at shorter distances • Observed neutron star (from analysis of bi- → matter harder to compress at high densities nary systems) are mostly around 1.5 M¯ PULSARS

• Modern calculations suggest that Mmax is probably Expected properties of neutron stars: less than 3M¯ and definitely less than 5M¯. See Phillips, “The Physics of Stars”, for an example cal- (a) Rotation period (c.f. white dwarfs, e.g. 40 Eri B, culation: incompressible star of constant density PWD = 1350 s) R m WD ' n ' From simple theory : 5/3 600 RNS 2 me conservation of angular momentum: with 2 MWD ∼ MNS and I = (2/5)MR (uniform sphere)

Ii i = If f

2 f = i(Ri/Rf ) −6 PNS ' 3 × 10 PWD ' 4 ms → neutron stars rotate rapidly when they form . but angular momentum is probably lost in the su- pernova explosion . rotation is likely to slow down rapidly (b) magnetic field . Flux conservation requires B dS = constant Z 2 2 Bi 4 ¡ Ri = Bf 4 ¡ Rf . Take largest observed white dwarf field, 4 BWD ' 5 × 10 T

2 10 BNS ' BWD(RWD/RNS) ' 10 T (upper limit) (c) Luminosity . neutron star forms at T ∼ 1011 K but T drops to ∼ 109 K within 1 day . main cooling process: neutrino emission (first ∼ 103 yr), then radiation 8 . after a few hundred years, Tinternal ∼ 10 K, Radio Pulsars: the P-B Diagram 6 Tsurface ∼ a few × 10 K. . star cools at constant R for ∼ 104 yr with ∼ 6 10 Tsurface 10 K young pulsars 2 4 26 ¡ L ∼ 4 R Ts ∼ 10 W (mostly X − rays, ¡ max ∼ 3 nm)

8 Discovery of neutron stars • The first was discovered by Bell and Hewish at old pulsars Cambridge in 1967 6 spin-up line . A radio interferometer (2048 dipole antennae) had been set up to study the scintillation which was ob- served when radio waves from distant point sources 4 Hubble line death line magnetic field: log B (T) B log field: magnetic passed through the solar wind. Bell discovered ms pulsars a signal, regularly spaced radio pulses 1.337 sec apart, coming from the same point in the sky every 2 night. 0.001 0.01 0.1 1.0 10 • Today, about 1500 pulsars are known spin period (sec) . Most have periods between 0.25 s and 2 s (average 0.8 s) . Extremely well defined pulse periods that challenge the best atomic clocks . Periods increase very gradually . spin-down timescale for young pulsars ∼ P/P˙ ∼ 106 − 107 yr The Crab Pulsar SUPERNOVA REMNANTS • The Crab nebula is the remnant of a supernova The Crab Nebula (plerionic/filled) explosion observed optically in 1054 AD. • The Crab pulsar is at the centre of the nebula, emit- VLT ting X-ray, optical and radio pulses with P = 0.033 s. • The Crab nebula is morphologically different from two other recent supernova remnants, Cas A and Tycho (both ∼ 400 yr old) which are shell-like. • The present rate of expansion of the nebula can be measured: uniform expansion extrapolates back to 90 years after the explosion, i.e. the expansion must be accelerating • The observed spectrum is a power law from ∼ 1014 Hz (IR) to ∼ 1 MeV (hard X-rays); also, in the extended nebulosity, the X-rays are 10-20 % polarised → signature of synchrotron radiation (relativistic Cassiopeia A (shell-like) electrons spiralling around magnetic field lines with Chandra B ∼ 10−7 T). (X-rays) • Synchrotron radiation today requires (i) replenish- ment of magnetic field and (i) continuous injection of energetic electrons. • Total power needed ∼ 5 × 1031 W • Energy source is a rotating neutron star (M ' 1.4 M¯, R ' 10 km)

2 2 2 ¡ U = (1/2) I = 2 I/P

2 dU 4 ¡ IP˙ = − dt P3 Taking I = (2/5) MR2 ∼ 1 × 1038 kg m2; P = 0.033 s;

P˙ = 4.2 × 10−13 → dU/dt ' 5 × 1031 W A simple pulsar model Pulsar dispersion measure • A pulsar can be modelled as a rapidly rotating neutron star with a strong dipole magnetic field in- • Consider an electromagnetic wave of the form E = E0cos (kx § wt) propagating through an ionised clined to the rotation axis at angle . medium where the number density of electrons is ne. • Pulsar emission is beamed (like a lighthouse beam) → observer has to be in the beam to see • The dispersion relation is pulsed emission. 2 2 2 2 2 1/2 = k c + p, p = (nee /(m ¢ 0)) (plasma frequency). rotational axis • Magnetic dipole radiation: • If < p → no wave propagation (e.g. ∼ few MHz cut- Magnetic Field

off to radio waves through the ionosphere). If > p, dUmag 2 ¡ 0 2 4 2

= − m sin 3 Ã ! propagation occurs with group velocity v given by: N dt 3c 4 ¡ g 4 2 2 −1/2 −1/2

¡ 2 2 ¡

32 m sin = − 0 d wp p neutron star 3 Ã ! 4 vg = = c 1 +  = c 1 +  3c 4 ¡ P 2 2 2 2 dk c k −    p  • taking dU/dt = −5 × 1031 W     27 2 • i.e. frequency dependent: longer wavelength has lower → m sin ' 4 × 10 A m . velocity. S • Hence, surface magnetic field • A pulse of radiation travels a distance l in time

¡ m charged partcile ' 0 ' 8 flux B 3 10 T. t = l/vg. 4 ¡ R • Frequency dependence of arrival time is given by: • Further argument for magnetic dipole radiation: £ £ t l £ vg l vg

= − ' − since vg ' c for >> p. dU d 2 2 £ £ £

rot 4 v c − = −I ∝ g dt dt 2 2 −3/2 2

£ v c c d g p p p 3 But = 1 +  ' . → = −C . 2 2 2 2 2 3 £

( − ) − dt p  p    2 3 £ ¤ £¤

1 1 1 ¡ ¢ − 1 4 mc 0 Integrate t =  2 2  Therefore = = −

2 £ £ 2C 0 t 2 ¡ t e n l   e 1 2 3 ¤ − − ¤ £ 16 1 4 ¡ mc ¢ 0 So t < with C = 3.5 × 10 s and = 190 s 2 Strictly = − since ne varies with l. 2C 2 £ t e nedl R • nedl is known as the DISPERSION MEASURE. It → t < 1250 yr (comparable to the known age ' 950 yr) Z is a useful distance indicator if ne is uniform (typical N.B. The physics underlying pulsar emission mechanisms value for the Galactic plane: 1 − 3 × 105 m−3) is very complicated and not well understood (Fruchter, Stinebring, Taylor 1988) MILLISECOND (RECYCLED) PULSARS radio • ∼ eclipse a group of 100 radio pulsars with very short spin pe- PSR 1957+20 riods (shortest: 1.6 ms) and relatively weak magnetic (black-widow fields (B ∼< 106 T) pulsar) • they are preferentially members of binary systems, •

secondary eclipse they have spin-down timescales comparable or longer time delay residual time delay than the Hubble time (age of the Universe)

0.10 0.15 0.20 0.25 0.30 0.35 0.40 • standard model orbital phase . these pulsars are neutron stars in binary systems that spin-down first, lose their strong magnetic field (due to accretion?) contact . and are spun-up by accretion from a companion discontinuity . magnetospheric accretion: magnetic field becomes Alfven surface . dominant when magnetic M pressure > ram pressure in . flow → flow follows magnetic 0.025 M . companion O M field lines (below rA) ablated wind . spin-up due to accretion of rAlfven shock angular momentum

γ • equilibrium spin period: vrot(rA) = vKepler(rA) 5 6/7 −3/7 → Peq =' 2 ms B/10 T M˙ /M˙ Edd B µ ¶ µ ¶ • a significant fraction of millisecond pulsars are single

pulsar → pulsar radiation has evaporated the companion wind radio . example: PSR 1957+20 (the black-widow pulsar): beam companion mass: only 0.025 M¯ (Phinney et al. . direct evidence for an evaporative wind from the millisecond 1988) radio eclipse (much larger than the secondary) pulsar → comet-like evaporative tail SCHWARZSCHILD BLACK HOLES

• event horizon: (after Michell 1784) . the escape velocity for a particle of mass m from 2GM v Orbits near Schwarzschild Black Holes an object of mass M and radius R is vesc = u u u R (11 km s−1 for , 600 km s−1 for Sun) t . assume photons have mass: m ∝ E (Newton’s corpuscular theory of light) photons . photons travel with the speed of light c

→ photons cannot escape, if vesc > c

2GM → R < R ≡ (Schwarzschild radius) s c2 . Rs = 3 km (M/ M¯) 2 3 4 6 rg Note: for neutron stars Rs ' 5 km; only a factor of 2 smaller than RNS → GR important • particle orbits near a black hole . the most tightly bound circular orbit has a radius 2 particles Rmin = 3 Rs = (6GM)/c (defines inner edge of accretion disk) . for a black hole accreting from a thin disk, the G M gravitational radius: r g = efficiency of energy generation is (usually) deter- c2 mined by the binding energy of the inner most sta- ble orbit (∼ 6 % for a Schwarzschild black hole) • no hair theorem: the structure of a black hole is com- pletely determined by its mass M, angular momentum L and electric charge Q