NEUTRON STARS. Maximum mass of a neutron star:

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1 NEUTRON STARS 193: Baade and Zwicky roosed that suernovae reresented the transition of normal stars to neutron stars 1939: Oenheimer and Volkoff ublished the first theoretical model 1967: discovery of ulsars by Bell and Hewish Maximum mass of a neutron star: Neutrons are fermions: degenerate neutrons are unable to suort a neutron star with a mass above a certain value (c.f. Chandrasekhar mass limit for white dwarfs). Imortant differences from white dwarf case: (i) interactions between neutrons are very imortant at high densities ii) very strong gravitational fields (i.e. use General Relativity) N.B. There is a maximum mass but the calculation is very difficult; the result is very imortant for black-hole searches in our Galaxy Crude estimate ignore interactions and General Relativity aly same theory as for white dwarfs: M max 6 M interactions between neutrons increase the theoretical maximum mass The interaction is attractive at distances 1. fm but reulsive at shorter distances matter harder to comress at high densities but, at high densities, degenerate neutrons are energetic enough to roduce new articles (hyerons and ions) the ressure is reduced because the new articles only roduce a small ressure Effect of gravity (gravitational binding energy of neutron star is comarable with rest mass): Gravity is strengthened at very high densities and ressures. Consider the ressure gradient: Einstein: dp dr = Gm r Newton: dp dr = Gm r (1 + P/( c))(1 + r3p/(mc)) 1 Gm/(rc ) Pressure P occurs on RHS. Increase of ressure, needed to oose gravitational collase, leads to strengthening of gravitational field need an equation of state P( n n interactions ) that takes account of Oenheimer and Volkoff (1939) calculated M max for a star comosed of non-interacting neutrons. Result: M max = 0.7 M Enhanced gravity leads to collase at finite density when neutrons are just becoming relativistic not ultrarelativistic Various calculations, using different comressibilities for neutron star matter, redict M max [1.5, 3] M R NS 7 15 km Observed neutron star masses (from analysis of binary systems) are mostly around 1.5 M

2 Ri = Bf Modern calculations suggest that M max is robably less than 3M and definitely less than 5M. See Phillis, The Physics of Stars, for an examle calculation: incomressible star of constant density PULSARS Exected roerties of neutron stars: (a) Rotation eriod (c.f. white dwarfs, e.g. 0 Eri B, P WD = 1350 s) R WD From simle theory : m n 600 R NS 5/3 m e conservation of angular momentum: with M WD M NS and I = (/5)MR (uniform shere) I i f = i = I f f i(r i /R f ) P NS P WD ms neutron stars rotate raidly when they form but angular momentum is robably lost in the suernova exlosion rotation is likely to slow down raidly (b) magnetic field Flux conservation requires B ds = constant B i R f Take largest observed white dwarf field, B WD 5 10 T B NS B WD (R WD /R NS ) T (uer limit)

3 magnetic field: log B (T) Radio Pulsars: the P-B Diagram sin-u line young ulsars Hubble line death line old ulsars ms ulsars sin eriod (sec) (c) Luminosity neutron star forms at T K but T dros to 10 9 K within 1 day main cooling rocess: neutrino emission (first 10 3 yr), then radiation after a few hundred years, T internal 10 8 K, T surface a few 10 6 K. star cools at constant R for 10 yr with T surface 10 6 K L R Ts 106 W (mostly X rays, Discovery of neutron stars max 3 nm) The first ulsar was discovered by Bell and Hewish at Cambridge in 1967 A radio interferometer (08 diole antennae) had been set u to study the scintillation which was observed when radio waves from distant oint sources assed through the solar wind. Bell discovered a signal, regularly saced radio ulses sec aart, coming from the same oint in the sky every night. Today, about 1500 ulsars are known Most have eriods between 0.5 s and s (average 0.8 s) Extremely well defined ulse eriods that challenge the best atomic clocks Periods increase very gradually sin-down timescale for young ulsars P/Ṗ yr

4 VLT Chandra (X-rays) SUPERNOVA REMNANTS The Crab Nebula (lerionic/filled) Cassioeia A (shell-like) The Crab Pulsar The Crab nebula is the remnant of a suernova exlosion observed otically in 105 AD. The Crab ulsar is at the centre of the nebula, emitting X-ray, otical and radio ulses with P = s. The Crab nebula is morhologically different from two other recent suernova remnants, Cas A and Tycho (both 00 yr old) which are shell-like. The resent rate of exansion of the nebula can be measured: uniform exansion extraolates back to 90 years after the exlosion, i.e. the exansion must be accelerating The observed sectrum is a ower law from 10 1 Hz (IR) to 1 MeV (hard X-rays); also, in the extended nebulosity, the X-rays are 10-0 % olarised signature of synchrotron radiation (relativistic electrons siralling around magnetic field lines with B 10 7 T). Synchrotron radiation today requires (i) relenishment of magnetic field and (i) continuous injection of energetic electrons. Total ower needed W Energy source is a rotating neutron star (M 1. M, R 10 km) U = (1/) I = I/P du dt = IṖ P 3 Taking I = (/5) MR kg m ; P = s; Ṗ = du/dt W

5 S A simle ulsar model A ulsar can be modelled as a raidly rotating neutron star with a strong diole magnetic field inclined to the rotation axis at angle. Pulsar emission is beamed (like a lighthouse beam) observer has to be in the beam to see ulsed emission. charged artcile flux rotational axis Magnetic Field N neutron star Magnetic diole radiation: du mag dt = ( 3c 3 = 3 3c 3 0 ( 0 ) m sin ) m sin P taking du/dt = W m sin 107 A m. Hence, surface magnetic field B 0m R T. Further argument for magnetic diole radiation: So t < 1 C du rot dt = I d dt d dt = C 3. Integrate t = 1 C with C = s and = 190 s 1 t < 150 yr (comarable to the known age 950 yr) N.B. The hysics underlying ulsar emission mechanisms is very comlicated and not well understood Pulsar disersion measure Consider an electromagnetic wave of the form E = E 0 cos (kx ± wt) roagating through an ionised medium where the number density of electrons is n e. The disersion relation is = k c +, = (nee/(m 0)) 1/ (lasma frequency). If < no wave roagation (e.g. few MHz cutoff to radio waves through the ionoshere). If >, roagation occurs with grou velocity v g given by: v g = d dk = c 1 + w c k 1/ = c 1 + 1/ i.e. frequency deendent: longer wavelength has lower velocity. A ulse of radiation travels a distance l in time t = l/v g. Frequency deendence of arrival time is given by: t But Strictly = l vg vg vg Therefore = ( t = l vg c since v g c for c 1 + ) 3/ >> c. 3 t = 1 3 t = mc 0 e n e l 3 mc 0 e since n e varies with l. n e dl n e dl is known as the DISPERSION MEASURE. It is a useful distance indicator if n e is uniform (tyical value for the Galactic lane: m 3 ).

6 time delay residual (Fruchter, Stinebring, Taylor 1988) radio eclise secondary eclise contact discontinuity orbital hase ablated wind M. comanion O shock PSR (black-widow ulsar) MILLISECOND (RECYCLED) PULSARS a grou of 100 radio ulsars with very short sin eriods (shortest: 1.6 ms) and relatively weak magnetic fields (B < 10 6 T) they are referentially members of binary systems, they have sin-down timescales comarable or longer than the Hubble time (age of the Universe) standard model these ulsars are neutron stars in binary systems that sin-down first, lose their strong magnetic field (due to accretion?) and are sun-u by accretion from a comanion Alfven surface.. M r Alfven M magnetosheric accretion: magnetic field becomes dominant when magnetic ressure > ram ressure in flow flow follows magnetic field lines (below r A ) sin-u due to accretion of angular momentum γ B equilibrium sin eriod: v rot (r A ) = v Keler (r A ) P eq = ms ( B/10 5 T ) 6/7 ( Ṁ/Ṁ Edd ) 3/7 a significant fraction of millisecond ulsars are single radio beam ulsar wind millisecond ulsar (Phinney et al. 1988) ulsar radiation has evaorated the comanion examle: PSR (the black-widow ulsar): comanion mass: only 0.05 M direct evidence for an evaorative wind from the radio eclise (much larger than the secondary) comet-like evaorative tail

7 SCHWARZSCHILD BLACK HOLES event horizon: (after Michell 178) the escae velocity for a article of mass m from an object of mass M and radius R is v esc = GM R (11 km s 1 for Earth, 600 km s 1 for Sun) assume hotons have mass: m E (Newton s coruscular theory of light) hotons travel with the seed of light c hotons cannot escae, if v esc > c Orbits near Schwarzschild Black Holes hotons R < R s GM c R s = 3 km (M/ M ) (Schwarzschild radius) Note: for neutron stars R s 5 km; only a factor of smaller than R NS GR imortant 3 6 rg article orbits near a black hole the most tightly bound circular orbit has a radius R min = 3 R s = (6GM)/c (defines inner edge of accretion disk) for a black hole accreting from a thin disk, the efficiency of energy generation is (usually) determined by the binding energy of the inner most stable orbit ( 6 % for a Schwarzschild black hole) articles gravitational radius: r = g G M c no hair theorem: the structure of a black hole is comletely determined by its mass M, angular momentum L and electric charge Q