Pulsars. D.Maino. Radio Astronomy II. Physics Dept., University of Milano. D.Maino Pulsars 1/43

Transcription

1 Pulsars D.Maino Physics Dept., University of Milano Radio Astronomy II D.Maino Pulsars 1/43

2 Pulsar Properties Pulsars are magnetized neutron stars emitting periodic, short pulses at radio wavelenghts with periods P between 1.4ms and 8.5s Pulsar=pulse and star. They show a lighthouse effect: they emit rotating beams of radiation flashing when beam sweeps line-of-sight Pulse periods are quite stable and P pulse = P rot Radio emission mechanism is still not completely clear. But pulsars are astrophysical tools for Neutron stars extreme conditions: deep grav. potential GM/(rc 2 ) 1, ρ g/cm 3, B Gauss Pulse periods fractional errors 10 6 power emitted in GW, NS masses, GR in strong fields, long-w GW from BH mergers or primordial. D.Maino Pulsars 2/43

3 Discovery Serendipitously in 1967 by graduate student Jocelyn Bell and Anthony Hewish Look for the un-expected! D.Maino Pulsars 3/43

4 Discovery Serendipitously in 1967 by graduate student Jocelyn Bell and Anthony Hewish Look for the un-expected! JB observed the dips with P 1.3s, different from any interference signal and reappear exactly once per sideral day origin outside the Solar System D.Maino Pulsars 4/43

5 A brief history : Fowler, Anderson & Stoner derived the Chandrasekhar limit for White Dwarf (WD) mass 1932: Chadwick discovers neutron 1934: Baade & Zwicky predict NSs (unobservable) 1967: LGM discovery by Jocelyn Bell (Nobel Prize to Hewish!) 1974: Hulse-Taylor binary pulsar - GW emission (Nobel Prize) today: 2500 known pulsars D.Maino Pulsars 5/43

6 Neutron Stars: Formation Neutron Stars: final stage of stellar evolution. Result from a core-collaspe SN 8M < progenitor mass < 20 30M Iron Core collapses when M > M Ch electron capture and photo-disintegration of iron nuclei T 10 9 K - e + p n + ν: neutrinos emission Core radius decreases from 3000 km down to 30 km; B flux and ω almost conserved ρ > g/cm 3 : neutron degeneracy pressure halts the collapse: shock on outer envolepose supernova GM 2 /R ergs liberated See Fogliazzo et al. PASA, 2015, 32, 9 D.Maino Pulsars 6/43

7 Why neutron stars? Short pulsation period very compact objects (WD, BH or NS) Stability of pulsation rule out BH due to their strong activity In normal pulsating stars: relation between P and ρ (but order of days not seconds) Consider a spherical star with M and R with angular velocity Ω = 2π/P: limit on P by centrifugal accel. at equator not exceeds gravitational accel. Ω 2 R < GM R 2 P 2 > ( 4πR 3 3 ) ( ) 3π 3π GM = ρ > 3π Gρ GP 2 CP1919 has P = 1.3s ρ 10 8 g/cm 3 consistent with WD. D.Maino Pulsars 7/43

8 Why neutron stars? Soon a pulsar with P = 0.033s discovered in the Crab Nebula ρ any stable WD (electron-degeneracy pressure) Crab Nebula is a SN remnant: observed by Chinese astronomers in 1054 This confirms Baade & Zwicky: NS are compact remnants of SN D.Maino Pulsars 8/43

9 NS & SuperNova Remnant The number of association between pulsar and its SNR is difficult: SNR are visible for upto 0.1Myr pulsars are 10 to 100 times longer lived High pulsar velocities ( 100km/s) compared to progenitors ( 1 10 km/s). Origin still debated: depends on SN Type and neutrinos also play a role. Best found in young SNR D.Maino Pulsars 9/43

10 Magnetic Field Most of normal stars possess a dipolar magnetic field 100G Star inner parts are fully ionized good conductor Remember:charges move along field lines and field lines are tied to charged particles Collapse: R 10 6 km 10km, flux Φ Φ = da B n is conserved (Alfven s Theo) field boosted by reaching (typically) G Additional dynamo effect can create larger fields ( G) observed in magnetars - very young neutron stars D.Maino Pulsars 10/43

11 Alfven s Theorem Consider a fluid (NS surface) and Maxwell Eqs. together with Ohm s Law B = 0 B t + E = 0 E + u B = J σ In the limit σ (matter is degenerate) we get B t = (u B) Consider B flux and its time derivative dφ B d B ds dt dt S D.Maino Pulsars 11/43

12 Alfven s Theorem Either B or ds are changing dφ B B = dt S t ds + B u dl C B = S t ds u B dl C [ ] B = (u B) ds = 0 t S Remember that a (b c) = b (c a) = c (a b) D.Maino Pulsars 12/43

13 Magnetic Dipole Pulsar Model Traditional magnetic dipole model (Pacini 67) Angle α between rotation and magn. axes Gaps where particles are accelerated Light-Cylinder: where co-roteting speed ΩR L = c D.Maino Pulsars 13/43

14 Magnetic Dipole Pulsar Model Inclined Magnetic Dipole emits e/m radiation with Ω = Ω rot extracting kinetic-energy from NS Larmor formula for a rotating electric dipole: P rad = 2 q2 v 2 3c 3 = 2 (q r sinα) 2 3 c 3 = 2 p 2 3 c 3 where p = qr is electric dipole moment and p = p sinα This is the same for the magnetic dipole m so that P rad = 2 3 m 2 c 3 D.Maino Pulsars 14/43

15 Magnetic Dipole Pulsar Model For a sphere of radius R with surface magnetic field B the moment m = BR 3 If dipole is rotating with velocity Ω it can be written as m = m 0 exp( iω t) and hence we get ṁ = iω m 0 exp( iω t) m = Ω 2 m 0 exp( iω t) = Ω 2 m Emitted power from dipole is therefore: P rad = 2 m 2 Ω4 3 c 3 = 2 ( B R 3 3c 3 sinα ) 2 where P is the pulsating period (Ω = 2π/P). ( ) 2π 4 P D.Maino Pulsars 15/43

16 Spin-Down Luminosity Rotational energy E of a spinning object is related to its moment of inertia I with E = I Ω 2 /2 = 2π 2 I /P 2 Moment of inertia of a sphere with radius R and mass M is I = 2/5 M R 2 and for a canonical NS it is around g cm 2. For the Crab Nebula pulsar with P = s the rotational energy E erg. The magnetic dipole extracts energy from NS and thus incresing the rotation period Ṗ > 0 Combining information on P and Ṗ we derive the rate Ė at which energy is changing We define spin-down luminosity as Ė de rot dt = d dt ( ) 1 2 I Ω2 = I Ω Ω D.Maino Pulsars 16/43

17 Spin-Down Luminosity The Crab pulsar has P = s and Ṗ = Assuming I = g cm 2 we get: Ė = 4π2 I Ṗ P 3 = 4π (0.033) erg s L If P rad Ė the Crab lumonosity at low-frequencies (ν = P 1 = 30Hz) is the entire radio output of our Galaxy! It is even large than the Eddington limit for a NS: ok since energy source is not accretion Most of this energy heats up the surrounding Crab Nebula: Megawave oven The observed bolometric luminosity is consisten with this simple model and with the following conversion and re-emission from radio to X-ray D.Maino Pulsars 17/43

18 Minimum Magnetic Field Strenght If P rad = Ė we can infer the minimum B strenght on the NS surface 2 ( BR 3 3c 3 sinα ) ( ) 2 2π 4 = 4π2 I Ṗ P P 3 B 2 3c 3 I PṖ = 2 4π 2 R 6 sin 2 α Since sin 2 α [0, 1] we can arrange terms to obtain B > ( 3c 3 ) 1/2 I ( ) 1/2 8π 2 R 6 PṖ The first term can be computed for a canonical pulsar ( ) ( ) 1/2 B > PṖ gauss s D.Maino Pulsars 18/43

19 Minimum Magnetic Field Strenght Let s again consider the Crab pulsar (P = 0.033s and Ṗ = ) ( ) B > ( ) 1/2 = gauss Extremely large B: the associated energy (U B = B 2 /8π 2 ) of 1 cm 3 of this B is erg 2 GW/year D.Maino Pulsars 19/43

20 Pulsar Age Suppose P rad = Ė and pulsar geometry almost constant (Bsinα constant) we can derive pulsar age τ from PṖ (initial P 0 actual period) PṖ constant PṖ = 8π2 R 6 B 2 sin 2 α 3c 3 I Rewrite PṖ = PṖ as PdP = PṖdt. Integrating over pulsar age: P P 0 PdP = since PṖ constant τ 0 (PṖ)dt = PṖ τ 0 dt D.Maino Pulsars 20/43

21 Pulsar Age Integration gives P 2 P0 2 = PṖτ 2 In the (reasonable) limit of P 0 P pulsar age is given by τ = P 2Ṗ Note that τ depends on the measured P and Ṗ but not on other pulsar characteristics e.g. radius R, momentum of inertia I or perperdicular magnetic field Bsinα Very young pulsars are very oblate (high rotational velocity) emits quadrupole GW radiation and hence slow down τ > true age for young pulsars D.Maino Pulsars 21/43

22 Pulsar Age Crab Pulsar: P = 0.033s and Ṗ = τ Crab = 0.033s s 1300yr somewhat larger since Crab Supernova dates 1054 AD Vela Pulsar: P = s and Ṗ = τ Vela = s 11300yr in agreement with other age estimates D.Maino Pulsars 22/43

23 PṖ diagram Plays a similar role of the HR diagram for normal stars Lines of constant τ, Ė,B Newly formed pulsars up-middle (SNR), move down-right to populate the 1 s pulsar. After that too slow to power radio emission Millisec pulsar down-left are mainly binary pulsars recycled via accretion D.Maino Pulsars 23/43

24 Braking Index If magnetic dipole radiation is the only source of spin-down then P rad = Ė that becomes 2 m 2 Ω4 3 c 3 = I Ω Ω Ω 3 Ω In general the braking index n is defined as Ω Ω n = CΩ n and for pure magnetic dipole we have n = 3 n is determined by the observed P,Ṗ and P D.Maino Pulsars 24/43

25 Braking Index Take time derivative of Ω ( ) CΩ Ω = CnΩ n 1 n Ω = n Ω Ω = n ( ) Ω Ω = n Ω 2 Ω Ω n = ΩΩ Ω 2 Convert from Ω to P Ω = 2πP 1 Ω = 2πP 2 Ṗ (Ṗ2 Ω = 2πP 2 P + 4πP 3 Ṗ 2 = 2π P 3 P ) P 2 D.Maino Pulsars 25/43

26 Braking Index Combining into n gives n = ( )( ) ( 2π P 4 2π P P 4π 2Ṗ2 P 2 n = 2 PP Ṗ 2 ) + 4πṖ2 P 3 n = 5 is quadrupole radiation (both e/m and GW) For different n other emission mechanisms accretion from debris disk after explosion instabilities on internal structure (glitches) a relativistic stellar wind D.Maino Pulsars 26/43

27 Glitches D.Maino Pulsars 27/43

28 Glitches Slow-down process is not constant but: almost periodic speed-up are present more common/regular in young pulsars and rare/spaced in older ones braking index during glitches n 1.4 no magnetic dipole Explanation involves internal NS structure Coupling of angular momenta of solid crust and superfluid interior Transfer of (quantized) angular momentum from superfluid to solid region impeded by crystal structure up to a certain point D.Maino Pulsars 28/43

29 Pulsar Binary Systems Almost all pulsars with P < 0.1s and B < G are in binary systems (variations in the observed pulse period) Left:nearly circular orbit (e = ) with a companion mass 0.47M. Right: Largest eccentricity (e = 0.888) and massive WD or NS companion Recycled Pulsars: they are restored despite their low B by accretion (mass and angular momentum) from companion NS B move hot ionized gas to polar gaps hot to emit in X-rays D.Maino Pulsars 29/43

30 Pulsars and ISM Pulsars short duration pulses, small sizes and high T B probes the ionized ISM D.Maino Pulsars 30/43

31 Pulsars and ISM Pulsars short duration pulses, small sizes and high T B probes the ionized ISM Electron in ISM form a cold plasma with refractive index µ = [ 1 ( νp ) ] 2 1/2 ν where ν is radio frequency and ν p is the plasma frequency ( e 2 n e v p = πm e ) 1/2 ( ne ) 1/ khz cm 3 where for typical ISM (n e 0.03 cm 3 ) v p 1.5 khz D.Maino Pulsars 31/43

32 Pulsars and ISM If µ is imaginary: no propagation through the ISM For propagating waves: µ < 1 and group velocity v g = µc For observations at ν ν p we get v g c ( 1 ν2 p 2ν 2 If the source is at distance d a delay due to dispersion is observed d dl t = d ( d v g c 1 + v ) p 2 0 2ν 2 dl d c 0 ) D.Maino Pulsars 32/43

33 Pulsars and ISM ( ) e 2 d t = ν 2 n e dl 2πm e c We define the dispersion measure - DM (units of pc cm 3 ) DM n e dl Final delay ( t ) sec ( DM pc cm 3 0 ) ( ν ) 2 MHz D.Maino Pulsars 33/43

34 Pulsars and ISM Gray-band: uncorrected ν 2 dispersion delay over the frequency band Data are folded into phase period With known DM (or after several trayes) we get the binned dispersion corrected pulse D.Maino Pulsars 34/43

35 Pulsars and ISM Inhomogeneities in turbolent ISM can cause diffractive and refractive scintillations similar to Earth atmosphere Diffractive: from minutes to hours time-scales and from khz to MHz frequency range with 1 order of magnitude flux variations Refractive: timescale of weeks and a < 2 factor on flux Scintillation is related to scattering and hence pulse broadering Inhomogeneities in ISM cause multiple scattering resulting in different waves path with strong ( ν 4 ) frequency dependence with a long exponential pulse tail D.Maino Pulsars 35/43

36 Pulsars and ISM D.Maino Pulsars 36/43

37 Pulsar Timing Pulsar Timing: the regular monitoring of rotation of the NS by measurements of the arrival of radio pulse times Once you have timing you can: probe the internal structure of NS precise astrometry test of theories of gravity in strong field regime possible detection of GWs Pulses are folded on the phase period increase S/N TOA (Time Of Arrival) precision can be very high µs Measured TOAs have to be corrected for seveal effects: t = t t t 0 + clock DM + R + E + S + R + E + S determined from accurete position/orbital meas. D.Maino Pulsars 37/43

38 Pulsar Timing For binary pulsars: time delay across NS orbit allows for high precision meas. of orbital parameters Keplerian parameters: projected semi-major axis x, longitude of periastron ω, time of periastron passage T 0, orbital period P b and eccentricity e Relativistic binaries are those involving a NS and another compact object i.e. WD, NS or even a BH Additional 5 post-kepleriar parameters rate of periastron advance ω (ellipt. orbits do not close in GR); orbital period decay P b (emission of GW) γ for time-dilation and grav. redshift and two other Shapiro terms (r and s called range and shape) D.Maino Pulsars 38/43

39 Pulsar Timing Ṗ b = 192π 5 ( Pb 2π ω = 3 γ = e ) 5/3 (T M) 2/3 (1 e 2 ) 1 2π ) 1/3 T 2/3 2π M 4/3 m 2(m 1 + m 2) ) ( Pb ( Pb ) 5/3 ( e e4 s = x r = T m 2 ( ) 2/3 Pb T 1/3 M 2/3 m 1 2 2π (1 e 2 ) 7/2 T 5/3 m1m2m 1/3 Here T GM /c 3 = µs is solar mass in time units, M = m 1 + m 2 total mass system in solar masses Given 2 of these we can derive system individual masses D.Maino Pulsars 39/43

40 Pulsar Timing Hulse and Taylor followed a binary pulsar B made by two NSs Accurate measurements of ω and γ allowed for the estation of the two masses m 1 and m 2 They compare the observed orbital period decay with GR predictions D.Maino Pulsars 40/43

41 Pulsar Timing D.Maino Pulsars 41/43

42 Pulsar Timing Array (PTA) Take a number of pulsars distributed across the sky GW background would manifest as a local (at Earth) distortions of TOA common to all pulsars The single i pulsar fraction frequency change δν i is δν i ν i = α i A(t) + N i (t) Cross-correlate fractional changes from pulsar i and pulsar j δ i δ j = α i α j A 2 (t) +α i A(t)N j (t) +α j A(t)N i (t) + N i (t)n j (t) All blue terms are 0 since GW amplitude is not correlated with intrinsic noise; same for individual noises D.Maino Pulsars 42/43

43 Overall GW detectors D.Maino Pulsars 43/43