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Transcript
Pulsars High Energy Astrophysics [email protected] http://www.mssl.ucl.ac.uk/ Introduction • Pulsars - isolated neutron stars - radiate energy via slowing down of rapid spinning motion (P usually < 1sec, dP/dt>0) • Pulsating X-ray sources / X-ray pulsators compact objects (generally neutron stars) in binary systems. Accrete matter from normal star companion. (P ~ 10s secs, dP/dt<0) Pulsars cont. • Discovered in radio Averaging over many pulses we see: Period pulse ~P/10 interpulse • Measuring pulse complicated by Doppler motion of Earth and frequency dispersion in pulse arrival times. • Individual pulses: av. very constant, individual pulses variable Pulsar period stability • Period extremely stable: 1 part in 10 12 indicates some mechanical clock mechanism - although this mechanism must be able to accommodate pulse variablity. • Pulsations of white dwarf??? (but Crab pulsar period (P~1/30 sec) too short). • Rotation of neutron star??? Rotation of a neutron star Gravitational force > centrifugal force GMm mv 2 r r 2 2r where v and P is the period P Reducing: M GM 4 r => 3 2 2 2 4r PG r P 2 M but 4 3 r 3 G = 6.67x10 -11 so 3 3 2 PG -1 -2 m kg s -3 ; PCrab = 33x10 s Substituting numbers for Crab then: 3 11 6 6.67 10 1100 10 kg m -3 so > 1.3 x 10 14 kg m -3 This is too high for a white dwarf (which has a density of ~ 10 9 kg m-3 ), so it must be a neutron star. Pulsar energetics • Pulsars slow down => lose rotational energy - can this account for observed emission? • Rotational energy: 1 2 I 4 2 I E I 2 2 2 2 P P 2 2 2 2 dE d 2 I 4 I dP so 2 3 dt dt P P dt Energetics - Crab pulsar Crab pulsar - M = 1 solar mass - P = 0.033 seconds - R = 10 4 m 2 2 2 30 8 2 I MR 2 10 10 kg m 5 5 = 0.8 x 10 38 kg m 2 dE 4 0.8 1038 1 dP 10 watts and 2 dt 0.033 P dt 1 dP 3 10 watts P dt 42 from observations: 1 dP P dt ~ 10 11 1 s thus energy lost dE 31 310 watts by the pulsar dt This rate of energy loss is comparable to that inferred from the observed emission, for example in the 2-20keV range, the observed luminosity in the Crab Nebula is approx. 1.5 x 10 30 watts. Thus the pulsar can power the nebula. Irregularities in pulsar emission • Short timescales - pulsar slow-down rate is remarkably uniform • Longer timescales - irregularities apparent - in particular, ‘glitches’ P glitch t A glitch is a discontinuous change of period Glitches Glitches are caused by stresses and fractures in the external layers, the so-called ‘crust’ of the neutron star. P 10 For example, ~ 10 P is the observed value for the Crab pulsar. Pulse profiles • Average pulse profile very uniform • Individual pulses/sub-pulses very different in shape, intensity and phase t Sub-pulses show high degree of polarization which changes throughout pulse envelope average envelope Neutron Stars • General parameters: - R ~ 10 km (104 m) - inner ~ 1018 kg m-3 = 1015g cm-3 - M ~ 0.2 - 3.2 solar masses - surface gravity ~ 1012 m s-2 • We are going to find magnetic induction, B, of a neutron star. Magnetic induction Magnetic flux, BdS constant surface radius Sun 8 4 Radius collapses from 7 x 10 m to 10 m Surface change gives Bns 7 10 4 BSun 10 8 2 9 5 10 The general field of Sun is uncertain but should be ~ 0.01 Tesla. Thus the field for the neutron star, Bns ~ 5 x 107 Tesla = 5 x 10 11 Gauss Next - how long does B ns last? Decay time of magnetic field Decay time of magnetic field: 3000m D s 0 10km 2 Polar cap D - typical dimension over which field varies significantly (for n.s., D ~ 3 x 10 3 m) s - conductivity Thus, t ~ (3 x 103 ) (1010 ) (410 -7) 11 3 ~ 10 seconds ~ 3 x 10 years But magnetic field Crab pulsar still intense after 1000 years => interior must be superconducting (s and both very large) Neutron stars very dense and zero-T energy supports star and prevents collapse. Neutron star structure crust Neutron star segment neutron 1. liquid solid Superfluid core? neutrons, 2. superconducting p+ and e1km crystallization of neutron 9km matter 10km 1018 kg m -3 Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice 2x1017 kg m -3 4.3x1014 kg m -3 109 kg m -3 1. Between densities of 4.3 x 10 14 kg m -3 and 2 x10 17 kg m -3, the lowest energy state is reached when nuclei are embedded in an electron and neutron fluid. 2. Above 2x1017 kg m -3, there is a continuous neutron fluid with electrons and protons as minor constituents. Pulsar Magnetosphere First, defining scale height p p h h h h p h The pressure difference supports the element of atmosphere The pressure difference is given by: p h h g h where is the density But p kT m thus (where m is the mass of constituent particles) p mg pm and p h kT kT Formula for scale height Integrating: mg p p0 exp h kT => pressure falls off exponentially with height in atmosphere with uniform temperature. kT has the dimensions of distance h0 mg and is called the ‘scale height’. Neutron star scale height For a neutron star, g ~ 1012 m s-2 T ~ 1 million K thus h0 ~ 0.01m Thus the atmosphere of a neutron star is only the order of 10cm! Forces exerted on particles Particle distribution determined by gravity etemperature Fg n s electromagnetism FB Gravity: Fgns me gns 9 10 31 10 10 12 18 Newton Magnetic force: FB evB 1.6 10 5 19 2 10 m 8 10 T 3 33 10 s 4 3 10 Newton This is a factor of 1013 larger than the gravitational force and thus dominates the particle distribution. Neutron star magnetosphere Neutron star rotating in vacuum: B Electric field induced immediately outside n.s. surface. E Bv 10 2 10 Vm 14 1 2 10 Vm 8 6 1 pd on scale of neutron star radius: ER 1018V Electron/proton expulsion Neutron star particle emission B electrons protons Cosmic rays In reality... • In reality, the charged particles will distribute themselves around the star to neutralize the electric field. => extensive magnetosphere forms • Number difference +ve and -ve charges: n n ~ 7 10 8 B Pperiod m 3 (B in Tesla) Crab pulsar particle density • This relationship gives an indication of the particle density n: • take, for example, the Crab pulsar 8 10 3 18 3 n 7 10 m 10 m 2 3 10 8 Pulsar models Magnetic and rotation axes co-aligned: eCo-rotating plasma, mag field lines are closed inside light cylinder Radius of light cylinder must satisfy: p light cylinder, R L 2RL c P A more realistic model... • Note that if radiation pulses are to be predicted, magnetic axis and rotation axis cannot be co-aligned. • => plasma distribution and magnetic field configuration around a neutron star is much more complicated. Radio beam Open magnetosphere B A better picture r=c/ Light cylinder Closed magnetosphere Neutron star mass = 1.4 solar masses radius = 10 km B = 10 4 to 109 Tesla The dipole aerial Even if a plasma is absent, a spinning neutron star will radiate if the magnetic and rotation axes do not coincide. This is the case of a a ‘dipole aerial’ dE 4 6 2 2 R B sin a dt Quick revision of pulsar structure 1. Pulsar can be thought of as a non-aligned rotating magnet. 2. Electromagnetic forces dominate over gravitational in magnetosphere. 3. Field lines which extend beyond the light cylinder are open. 4. Particles escape along open field lines, accelerated by strong electric fields. Radiation mechanisms in pulsars Emission mechanisms Total radiation intensity exceeds does not exceed Summed intensity of spontaneous radiation of individual particles coherent incoherent Incoherent emission - example eg. Radiating particles in thermodynamical equilibrium ie thermal emission. blackbody => max emissivity So is pulsar emission thermal? consider radio: n~10 8 Hz; 100MHz; 3m Use Rayleigh-Jean approximation to find T: 2kTn I n 2 c 2 Watts m -2 Hz -1ster -1 Flux density at Earth, F~10-25 watts m -2 Hz -1 Source radius, R~10km at distance D~1kpc then: D 10 3 10 F I F 2 2 4 10 R (1) 2 25 19 2 6 = 10 watts m -2 Hz -1 ster -1 From equation (1): I n c 10 3 10 T K 2 23 8 2kn 2 1.4 10 10 2 310 K 29 6 8 2 2 this is much higher than a radio blackbody temperature K Incoherent X-ray emission? • In some pulsars, eg. Crab, there are also pulses at IR, optical, X-rays and g-rays. • - Are these also coherent? • Probably not – brightness temperature of Xrays is about 100 billion K, equivalent to electron energies 10MeV, so consistent with incoherent emission. radio coherent IR, optical, X-rays, g-rays incoherent Models of Coherent Emission high-B sets up large pd => high-E particles e- ep+ electron-positron pair cascade B=1e8Tesla 1e16V cascades results in bunches of particles which can radiate coherently in sheets Emission processes in pulsars • Important processes in magnetic fields : - cyclotron Optical & X-ray emission in pulsars - synchrotron • Curvature radiation => radio emission B V. high mag fields; efollow field lines very closely, pitch angle ~ 0 Curvature Radiation • This is similar to synchrotron radiation. If ve- ~ c and = radius of curvature, radiation v. similar to e- in circular orbit with: c where nL is the L 2g gyrofrequency ‘effective frequency’ of emission is given by: m Lg 3 Curvature vs Synchrotron Synchrotron Curvature B B • Spectrum of curvature radiation - similar to synchrotron radiation, Flux n 1/3 e -n n nm • e- intensity c.r. << cyclotron or synchrotron => radio produced this way, need coherence X-rays from curvature radiation? • At frequency 1018Hz luminosity ~ 10 29 J/s requires g ~ 105 and no. particles radiating nV ~ 10 40- 1041 depending on density. • This is too many for such energetic particles => X-rays emitted by normal synchrotron Beaming of pulsar radiation • Beaming => radiation highly directional • Take into account - radio coherent, X-rays incoherent - location radiation source dep on frequency • Model: - radio from magnetic poles - X-rays from light cylinder Radio beam Open magnetosphere B A better picture r=c/ Light cylinder Closed magnetosphere Neutron star mass = 1.4 solar masses radius = 10 km B = 10 4 to 109 Tesla Magnetic poles Radiation source localized near mag poles. b (simple, axisymmetric case) Rad source localized near poles, narrow beam produced along mag field. Polar caps defined by field lines tangential to light cylinder. 2 b ~ 1 10 light cylinder Important observed properties • Pulses observed only when beam points at Earth. • Rad source probably localized within light cylinder close to neutron star surface… - no ‘wandering’ and directionality • Problem: ALL radiation mechanisms at different frequencies (coherent or not) must have same orientation along magnetic field. Origin of subpulses subpulses Brightening on boundaries between closed and open lines may produce subpulses boundary co-rotating plasma Light Cylinder Radiation source close to surface of light cylinder. P P’ simplified case Light cylinder – realistic but complex! Top view aligned and B Cross-section torus aligned and B r=c/ B What we see? • Relativistic beaming may be caused by ~c motion of source near light cylinder radiation concentrated into beam width : g , 1 g 1 1 b 2 (the Lorentz factor) • Also effect due to time compression (2g 2 ), so beam sweeps across observer in time: P 2 P 1 2 3 2g g 4g Long Period Pulsars • Not generally seen in optical or X-rays - is this emission produced at light cylinder? power radiated by synchrotron E B For dipole magnetic field : Also : cP RL 2 2 2 Br 3 • So if particles of the necessary energy E exist in all pulsars and emission occurs at RL , we expect: 6 - radiated power P • and thus long period pulsars are weak emitters. In summary... • Radio emission - coherent - curvature radiation at polar caps • X-ray emission - incoherent - synchrotron radiation at light cylinder Magnetic energy & nebula • Neutron star slows down => energy sufficient to feed nebula • What about the magnetic energy? Consider energy released at light 2RL cylinder. RL Area = 2 4RL • Magnetic field at RL is stretched out to v~c. • Magnetic energy density = B2 2 0 • Mag energy crossing light cylinder per sec: 3 B2 R0 2 But for PB 4RL c BRL B0 3 2 0 RL mag dipole 6 so 2 R 2 cR 2 L PB B0 0 RL 0 Substituting values for the Crab pulsar: PB 10 8 2 10 Js 31 6 10 2 3 10 10 6 7 4 10 10 4 8 6 2 Js 1 1 like rotational energy release, this is also comparable to observed emission from Crab Nebula Age of Pulsars . Ratio P / P (time) is known as ‘age’ of pulsar In reality, may be longer than the real age. Pulsar characteristic lifetime ~ 107 years Total no observable pulsars ~ 5 x 10 4 Pulsar Population • To sustain this population then, 1 pulsar must form every 50 years. • cf SN rate of 1 every 50-100 years • only 8 pulsars associated with visible SNRs (pulsar lifetime 1-10million years, SNRs 10-100 thousand... so consistent) • but not all SN may produce pulsars!!!