NEUTRON STARS . 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 Volko® published the ¯rst ² theoretical model E®ect of gravity (gravitational binding energy of neu- tron star is comparable with rest mass): 1967: discovery of pulsars by Bell and Hewish ² Gravity is strengthened at very high densities and Maximum mass of a neutron star: pressures. Consider the pressure gradient: ² 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) ² Important di®erences from white dwarf case: ² 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 ¯eld ii) very strong gravitational ¯elds (i.e. use General ² need an equation of state P( ) that takes account of Relativity) n ¡ n interactions N.B. There is a maximum mass but the calculation ² Oppenheimer and Volko® (1939) calculated M for is very di±cult; 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¯ ² Crude estimate{ ignore interactions and General Rel- ² Enhanced gravity leads to collapse at ¯nite density ativity { apply same theory as for white dwarfs: when neutrons are just becoming relativistic { not ul- Mmax ' 6 M¯ trarelativistic ² interactions between neutrons increase the theoretical ² Various calculations, using di®erent compressibilities maximum mass for neutron star matter, predict » . The interaction is attractive at distances 1:4 fm Mmax 2 [1:5; 3] M¯ RNS ' 7 ¡ 15 km but repulsive at shorter distances ² Observed neutron star masses (from analysis of bi- ! matter harder to compress at high densities nary systems) are mostly around 1:5 M¯ PULSARS ² Modern calculations suggest that Mmax is probably Expected properties of neutron stars: less than 3M¯ and de¯nitely 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 ¯eld . Flux conservation requires B dS = constant Z 2 2 Bi 4 ¡ Ri = Bf 4 ¡ Rf . Take largest observed white dwarf ¯eld, 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 (¯rst » 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 ² The ¯rst pulsar 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) 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 ² 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 de¯ned 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 ² The Crab nebula is the remnant of a supernova The Crab Nebula (plerionic/filled) explosion observed optically in 1054 AD. ² The Crab pulsar is at the centre of the nebula, emit- VLT ting X-ray, optical and radio pulses with P = 0:033 s. ² The Crab nebula is morphologically di®erent from two other recent supernova remnants, Cas A and Tycho (both » 400 yr old) which are shell-like. ² 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 ² 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 ¯eld lines with Chandra B » 10¡7 T). (X-rays) ² Synchrotron radiation today requires (i) replenish- ment of magnetic ¯eld and (i) continuous injection of energetic electrons. ² Total power needed » 5 £ 1031 W ² 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 ² A pulsar can be modelled as a rapidly rotating neutron star with a strong dipole magnetic ¯eld in- ² 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. ² Pulsar emission is beamed (like a lighthouse beam) ! observer has to be in the beam to see ² The dispersion relation is pulsed emission. 2 2 2 2 2 1=2 = k c + p; p = (nee =(m ¢ 0)) (plasma frequency): rotational axis ² Magnetic dipole radiation: ² If < p ! no wave propagation (e.g. » few MHz cut- Magnetic Field o® to radio waves through the ionosphere). If > p, dUmag 2 ¡ 0 2 4 2 = ¡ m sin 3 à ! 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 à ! 4 vg = = c 01 + 1 = c 01 + 1 3c 4 ¡ P 2 2 2 2 dk c k ¡ B C B p C ² taking dU=dt = ¡5 £ 1031 W @ A @ A 27 2 ² i.e. frequency dependent: longer wavelength has lower ! m sin ' 4 £ 10 A m . velocity. S ² Hence, surface magnetic ¯eld ² 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 ² Frequency dependence of arrival time is given by: ² 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 = 01 + 1 ' : ! = ¡C : 2 2 2 2 2 3 £ ( ¡ ) ¡ dt p B p C @ A 2 3 £ ¤ £¤ 1 1 1 ¡ ¢ ¡ 1 4 mc 0 Integrate t = 0 2 2 1 Therefore = = ¡ 2 £ £ 2C 0 t 2 ¡ t e n l @ A 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 ² 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 ² » 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 ¯elds (B »< 106 T) pulsar) ² they are preferentially members of binary systems, ² secondary eclipse they have spin-down timescales comparable or longer time delay residual than the Hubble time (age of the Universe) 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ² standard model orbital phase . these pulsars are neutron stars in binary systems that spin-down ¯rst, lose their strong magnetic ¯eld (due to accretion?) contact . and are spun-up by accretion from a companion discontinuity . magnetospheric accretion: magnetic ¯eld becomes Alfven surface . dominant when magnetic M pressure > ram pressure in . flow ! flow follows magnetic 0.025 M . companion O M ¯eld lines (below rA) ablated wind . spin-up due to accretion of rAlfven shock angular momentum γ ² equilibrium spin period: vrot(rA) = vKepler(rA) 5 6=7 ¡3=7 ! Peq =' 2 ms B=10 T M_ =M_ Edd B µ ¶ µ ¶ ² a signi¯cant fraction of millisecond pulsars are single pulsar ! pulsar radiation has evaporated the companion wind radio .