Scuola Nazionale di Astrofisica Oggetti Compatti e Pulsar Scienza con ALMA (Cagliari, 20 26 Maggio 2007) Accretion Discs around a Magnetized Neutron Star Scuola Nazionale di Astrofisica Oggetti Compatti e Pulsar Scienza con ALMA (Cagliari, 20 26 Maggio 2007) Accretion Discs around a Magnetized Neutron Star L.Burderi Universit degli Studi di Cagliari Dipartimento di Fisica SP Monserratu- Sestu KM 0.7, Monserrato Italy TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AA A A A A Spin-up in XTE J1807: a 5ms accreting pulsar in a LMXB Riggio et al. 2007 Spin down in the case of XTE J Spin down in XTE J0929, the slowest (5.4 ms) among accreting MSPs. During the only outburst of this source observed by RXTE. Measured spin-down rate: dot = -5.5 x Hz/s Porb = 44 min s = 185 Hz Disc Magnetic Field Interaction. Rm = 10 B 8 4/7 dM/dt -8 -2/7 m 1/7 km Ram Pressure proportional to dM/dt R -5/2 (Disc Pressure proportional to dM/dt 17/20 ) Magnetic Pressure Proportional to B 2 R -6 Radial accretion onto magnetized NS Naive determination of R A (Alfvn radius): inside R A the magnetic field control the accretion flow HP: radial flow P MAG = P RAM with and and Disk accretion onto magnetized NS (Ghosh & Lamb) z R j Existence of a magnetosphere of radius Rm just inside Rm: B Z (Rm) = 2 B Z NS (Rm) Outside Rm: B Z (R > Rm) = 0 Sheet current induced by the Lorentz force v R = v ff Total screening (because of low resistivity of the plasma ). Location of the inner radius: Pmag = Pdisc gives: Rm = R A (with = ) Accretion conditions (Illarionov & Sunyaev 1975) Accretion regime Rm < R CO < R LC Pulsar spin-up Accretion of matter onto NS (magnetic poles) Energy release L = dM/dt (G M/R NS ) Accretion of angular momentum dJ/dt = +j dM/dt j = (G M Rm) 1/2 is the specific angular momentum at Rm R co = 15 P 3 2/3 m 1/3 km R LC = 47.7 P 3 km Propeller conditions (Illarionov & Sunyaev 1975) Propeller regime R CO < Rm < R LC Pulsar spin-down Centrifugal force > Gravitational force Accretion is centrifugally inhibited Ejection of matter and angular momentum dJ/dt = -( NS Rm 2 j) dM/dt j = (G M Rm) 1/2 is the specific angular momentum at Rm R co = 15 P 3 2/3 m 1/3 km R LC = 47.7 P 3 km Problems in Ghosh and Lamb approach Perfect diamagnetic disc: no accretion beyond R m ! Solution: Plasma penetrates through the magnetosphere via Rayleigh-Taylor & Kelvin-Helmholtz instabilities. The onset of these instabilities depends on several factors (e.g. Cooling of the plasma for RT or |V K (Rm) V B (Rm)| for KH) not related to the disc structure Consequences: Accretion is not stationary (contrarily to what is observed) (However stationary accretion could be restored if RT rate is regulated by IRRADIATION of the plasma by accretion luminosity) Torque on the NS in diamagnetic discs Matter accretes through a Keplerian disk truncated at magnetospheric radius Rm proportional to (dM/dt) - In standard disk accretion = 2/7 Because of the spin frequency increases as + dot t with dot = /I and the pulse phase delays decreases as = 0 - dot t 2 Difficulty: during accretion always spin-up (contrarily to what is observed in XTE J0929 and XTE J1814) Matter transfers to the neutron star its specific angular momentum j= (GM Rm) 1/2 at Rm, causing a torque = j dM/dt Threaded disc model Bz BB is the screening factor Dragging of the field line: the B component is generated B is amplificated by differential rotation In this model the amplification of the field is limited by turbulent diffusion (Wang 1995) Threaded discs The plasma is not a super-conductor: diffusion of the magnetic field in the disc Threaded disc prescription for the magnetic field (threading induced by turbulent diffusion) and is the screening factor is the B amplification factor ( is the Shakura Sunyaev parametrization of the viscosity) The torque on a disc anulus of width dR exerted by the magnetic field is: The torque is positive for R Rco Spin up and spin-down during accretion are both possible! Torques on the NS in threaded discs 1) 1) Material torque: 2) Spin-up magnetic torque: 3) Spin-down magnetic torque: where R 0 is the inner disc radius Key problem of the Threaded Disc model A standard Shakura Sunyaev accretion disc is NOT threaded! Zero order approach : the disc is just perturbed by the magnetic field (inconsistent with the propeller regime!) No prescriptions for R 0 in this case (but see below) thus Computation of the total torque on the NS IMPOSSIBLE! Speculatons by several authors (e.g. Rappaport et al. 2004, see also works of Andersson 2006, Matthews 2005, 2004, Wang 1995) that spin-down during accretion IS POSSIBLE (consistent with timing on XTE J0929 and XTE J1814 ) A self consistent solution of the TD is required! Viscosity and AM transport in a Keplerian flow v R R R + /2 R - /2 Radial Collision Tangential Collision m Viscosity and AM transport in a Keplerian flow 1) Radial collisions: radial impact inward = L 1 radial impact outward = L 2 Radial collisions balance: L radial = L 1 - L 2 L 1 > L 2 -> inward transport of AM 2) Tangential collisions: tangential impact inward = L 3 tangential impact outward = L 4 Tangential collisions balance: L tangential = L 3 - L 4 L 3 outward transport of AM Overall AM balance: L total = L radial + L tangential | L tangential | > | L radial | -> L total Rco The spin-up problem The spin-up problem For ANY value of R A and Rco the matter accretes onto the NS, the NS gains angular momentum and spins-up As the spin up process goes on, Rco goes to 0 while R 0 0 Actually, R 0 stops at the NS surface Thus the spin limit is the centrifugal limit at the NS surface. The propeller problem The propeller problem A sudden drop of Mdot causes: R0 => Rco (from inside) The propeller is NOT possible! (in strong conflict with the naive argument of the centrifugal barrier) but... for k >> 0 two other solutions exists! Correct computation of R 0 in a TD In general we have three possibilities: 1) -> the disc is an unperturbed SS accretion disc 2) MAG negative and -> the magnetic torques extract more angular momentum than needed to keep the matter orbiting in a quasi Keplerian way: the disc is truncated and the matter accretes in almost free-fall 3) MAG positive and -> the magnetic torques give to the matter more angular momentum than needed to stay in orbit -> the matter is expelled (PROPELLER) Truncation radius in a slow rotator Truncation radius in a slow rotator Truncation radii in a fast rotator Truncation radii in a fast rotator Torques in a TH disc Spin evolution of a magnetized neutron star Spin evolution of a magnetized neutron star Truncation radii in a TH disc Comparison of a SS and a TH disc Torques on a NS with a TH disc The torque on the NS is always positive for accretors (although almost zero close to the spin equilibrium line).Timing results shows that two accreting millisecond X-ray pulsars are spinning down with almost constant spin derivative (despite a factor 10 decrease in the accretion rate!). Thus: 1) Magnetodipole radiation mechanism is effective both in radio-pulsar and in accreting weakly magnetized NS, strongly supporting the recycling scenario 2) the accretion torque is almost zero (the systems are almost at equilibrium) Conclusions: New analytical solutions for accretion discs around magnetized NS derived The disc exists for ANY value of the magnetic field and NS spin For R A > 1.58 Rco (new correct spin equilibrium relation neglecting the magnetodipole radiation) a propeller sets-in: The truncation radius jumps from 0.88 Rco to 1.50 Rco A cavity forms around the NS. Its radius roughly scales as ( R A /Rco) 3/4 A standard SS accretion disc surrounds this cavity The incoming matter is ejected with the escape speed from a boundary layer extending inward for about 30% of the cavity radius The torque on the NS is always positive for accretors (although almost zero close to the spin equilibrium line) Timing results shows that two accreting millisecond X-ray pulsars are spinning down indicating that 1) the accretion torque is almost zero (the systems are almost at equilibrium) 2) Magnetodipole radiation mechanism is effective both in radio-pulsar and in accreting weakly magnetized NS, strongly supporting the recycling scenario Thats all Folks! Shakura-Sunyaev discs AM balance in a disc anulus of width dR SS discs: Viscous torque on the anulus = torque exerted ON the ANULUS at the INNER BOUNDARY torque exerted BY the ANULUS at the OUTER BOUNDARY Outward AM flow (Keplerian motion plus inward flow of matter) TH discs: Viscous torque + Magnetic torque = Outward AM flow vv R R + /2 R - /2