Chapter 6 Accretion onto Compact Stars

Transcription

1 Chapter 6 Accretion onto Compact Stars General picture of the inner accretion flow around a NS/WD 1

2 1. Boundary layers around nonmagnetic white dwarfs Luminosity For non- or weakly-magnetized accreting stars, the accretion disks can extend down to the surface of the stars, with a boundary layer of radial extent b lying between the disk and the star, where the angular velocity of accreting material must decrease from Ω(R +b) = Ω K (R +b) to the surface angular velocity Ω.

3 The luminosity radiated from the boundary layer is given by 1 L = M& BL [( R* + b) ΩK ( R* + b) R* Ω* ] G* Ω* where the last term on rhs is the work acted by the viscous torque to spin up the star given by 3 G = π R νσω ' = M& ( R* ΩK R Ω) and G MR & ( Ω ) * * = * K Ω Finally we get the boundary layer luminosity GMM& Ω* LBL = [1 ] R Ω ( R ) * K * 3

4 Size and temperature The radial component of the Euler equation is v v R φ 1 P vk vr + + = 0 R R ρ R R Since v φ <v K, and v R <c s, the equation in the boundary layer is approximately c s GM ~ R b * or R* H b ~ cs ~ << H << R* GM R * 4

5 However, detailed calculations show that the radiation emitted by the boundary layer can emerge through a region of radial extent ~H on the two faces of the disk. 5 (Adopted from Popham & Sunyaev 001, ApJ, 547, 355)

6 In optically thick case the characteristic blackbody temperature is 4 GMM& 4πR* HσTBL ~ R* or 5 7/3 11/3 5/3 T ~1 10 M& M R K BL This means that some CVs ought to be soft X-ray sources, although most of the black body emission ( 0.1keV) is hidden by interstellar absorption. 6

7 If the accretion rate is sufficiently low and the boundary layer is optically thin, it could be much hotter and emits hard (~ 10 kev) X-rays, as observed in some CVs. There are two types of hard X-ray production mechanism proposed. (1) If the circulating gas with highly supersonic rotating velocity near the white dwarf surface is strongly shocked, the shocked temperature can be as high as 3 μmhgm 8 1 Ts = M1R9 K 16 kr * 7

8 () The optically thin boundary layer is likely to be thermally unstable to turbulent viscous heating, rapidly achieving hard X-ray temperature (~10 8 K), the gas is likely to expand out of the boundary layer and form a hard X-ray corona around most of the white dwarf. 8

9 . Accretion onto magnetized neutron stars and white dwarfs Neutron stars and white dwarfs often possess magnetic fields strong enough to disrupt the disk flow. In quasi-spherical case the magnetic pressure will begin to control the matter flow, and thus disrupt the spherically symmetric infall at the magnetospheric radius R M, where it first exceeds the ram and gas pressure of the matter, i.e. μ = ρv ff 6 R 8πR M M for dipolar fields, 9

10 or R = 5 10 M& M μ cm M 8 /7 1/7 4/ The inner radius R 0 of accretion disks is also close to R M. Inside the magnetosphere the accretion flow is channeled along the field lines onto a small fraction of the stellar surface. In polar coordinates (r,θ) with origin at the star s center, the dipole geometry follows r=csin θ, where C is a constant for a particular field line. At the inner edge of the disk we have r=r M and θ =α. 10

11 This field line crosses the stellar surface at R=R and θ=β given by sin β = ( R / R )sin α * M So accretion cannot take place outside a pole cap of half-angle β. The area of the two accreting pole caps is a fraction πr* sin β R* sin α R* fdisk ~ ~ 4πR* RM RM of the stellar surface. Generally 4 1 f ~ (10 disk 10 ). 11

12 The Ghosh & Lamb (1979a, b) model for disk-field interaction In this model the stellar magnetic field is assumed to penetrate the disk, and become twisted because of the differential rotation between the star and the disk. The total torque N exerted on a star contains two components, N= N 0 + N mag where N = MR & Ω ( ) K R

13 The magnetic torque N mag results from the azimuthal field component Bφ, which is generated by shear motion between the disk and the vertical field component B z, B B ( 1 ω) φ and z Nmag = R B B z dr R φ 0 The torque N mag can be positive and negative, depending on the value of the fastness parameter ω 3/7 1 */ ( R K 0) ( PM =Ω Ω & ), 13

14 For slow rotators with ω <<1, disk accretion leads to increase of the stellar angular momentum (spin-up). For fast rotators (ω 1), the interaction of the stellar magnetic field with parts of the disk at R > R 0, so that N<0, and the star spins down. N/N 0 0 ω cr 1 ω 14

15 The balance of the spin-up and spin-down torques leads to an equilibrium period (at which N=0) given by the condition ω = ω cr , or 1 /7 3/7 6/7 3/7 P = 3 ω M R μ L s eq cr

16 3. Accretion columns of white dwarfs Accretion flow around magnetized stars is channeled onto the polar caps. The resulting configuration near the stellar surface is known as accretion column. Above the polar caps the accreting matter is in free fall and highly supersonic. In order to accrete, the infalling material must be decelerated to subsonic velocities, and some sort of strong shock must occur in the accretion stream. 16

17 After the shock the kinetic energy (most is brought by ions) is randomized and turned into thermal energy. It is of vital importance to see where the ions are stopped (below or above the photosphere). We first need to calculate the optical depth τ =κ R ρ b λ s, where the stopping length of an ion is 3/4 3/4 7/4 1/4 λ s 3 M & f M R cm From Eq. (3.36) in FKR

18 Assume that the photosphere is a small fraction (~f ) of the white dwarf surface, at the base of the column the temperature is Lacc 1/4 5 1/4 1/4 1/4 3/4 Tb = ( ) = 1 10 M& 16 f M1 R9 K 4πR fσ. * The density in the white dwarf envelope where stopping occurs is μmh μmh μmh M& ρb = Pb = Pram = ( )( v ) ff kt kt kt 4πR f b b b * 18

19 The optical depth / 8 11/ 8 5 / 8 τ = κ / / Rρbλs = λs λph M& 16 f M1 R9 << 1 Where for Kramers opacity, the mean free path of a photon with frequency ~kt b /h is 1 3 1/8 1/8 3/8 7/8 λph = 7 10 M& 16 f M1 R9 cm κ ρ R b So the accreting matter must be shocked and decelerated above the photosphere. 19

20 In steady state, the accretion energy released below the shock must be removed from the post-shock column at the same rate at which it is deposited, by a combination of cooling mechanism such as radiation and particle transport processes. In order to construct an accretion column we must specify how the shock-heated material cools. If the cooling were inadequate the post-shock gas would expand adiabatically and raise the shock; if cooling became too effective the shock would move in towards the surface, reducing the volume of the cooling region. 0

21 The post-shock velocity v =v ff /4, the post-shock density is then / 3/ ρ = M& /(4 πr fv) M& 16f M1 R9 gcm, corresponding to an electron density / 3/ 3 N = M& f M R cm The cylindrical radius of the column is approximately 1/ 8 1/ d = f R = 10 f R9 cm. 1

22 8 At temperature 10 K, the major opacity source is electron scattering. Thus the horizontal optical depth is 1 1/ 1/ τ es ( d) ~ Neσ d 8 10 M& T 16 f M1 R9 so that radiative cooling will be optically thin. Thus all this radiation will leave through the sides of the column after at most a few scatterings.

23 At high temperature T s 10 8 K, radiative cooling is dominated by free-free (bremsstrahlung) emission, and the characteristic cooling time is 3NkT e 1 trad ~ 7 M& 16 f MR 1 9 s 4π j br During this time the gas will be settling from the shock with a distance 8 1 3/ 1/ Drad ~ vtrad 9 10 M& 16 f M1 R9 cm before cooling to a temperature of the order of the photospheric value T b. So it gives an estimate of the shock height when radiation alone cools the post-shock gas. 3

24 The equations of continuity, momentum and energy for a steady accretion flow in the column (in one dimension since the shock height D<R ) are ρ v = constant dv dp ρv + + gρ = 0 dz dz d 3kT v ρvkt [ ρv( + + gz) + ] = 4πjbr dz μmh μmh where z = 0 is the base of the column (v ~ 0). 4

25 The solution is T z / 5 = ( ) Ts D with D ~ D rad /3. We also have v v = T / T / = ρ / ρ 5

26 In more realistic case the accretion flow above the white dwarf is not homogeneous, but could be broken up into blobs. 6

27 Sufficient dense blobs can penetrate the photosphere and radiate almost all their energy as soft X-rays. This may explain why AM Her systems show large soft X-ray excess. 7

28 4. Energy Extraction From Rotating Black Holes Efficiency of energy extraction In Newtonian theory, the energy equation for a particle orbiting a mass M is dr ( ) dt 1 + V ( r) = E where E = constant is the total energy of the particle per unit mass and V ( r) h r = GM r is the effective potential for the particle of angular momentum h per unit mass. 8

29 In general relativity the energy equation changes to be (the Schwarzschild solution) 1 c dr ds ( ) + V ( r) = E where s is the proper time and related with the coordinate (absolute) time t by dt d s = 1 E R S / r where R S = GM/c is the Schwarzschild radius. 9

30 The effective potential is given by V ( r) = (1 RS / r)(1 + h / c r ), which is shown in the figure for various values of the dimensionless specific angular momentum. The dot-dashed curve in the figure is the pseudo-newtonian potential V ( r) = r 1 R S which can be used to mimic qualitatively the effects of space-time curvature on an orbiting body in a Newtonian description. 30

31 In both cases the orbital motion is possible only for V(r) E However, V(r) in GR has some distinct features. (1) If h < 3GM / c, there is no turning point. () If h 3GM / c, V(r) has a maximum and minimum Circular orbits where dr/ds = 0 are possible at radii such that V/ r =0. This now includes the maxima and minima. Since stable orbits require V / r > 0 only the latter is possible. 31

32 This gives r = ( R / 4)[ H ms S + ( H where H=c h/gm, if 4 1H ) 1/ h 3GM / c. ] The innermost of the stable circular orbits occurs at r ms = 3R S. The efficiency of energy generation is GMm /rms 1 mc 1 3

33 A rotating black hole is characterized by a mass M and angular momentum per unit mass J, or in dimensionless form m = GM/c and a = J/c respectively. The horizon of a Kerr black hole is r h = m + (m a ) 1/. The effective potential can be defined as the minimum value of the energy per unit mass E min for which motion is possible at each radius r, V() r = E min 1/ 1/ 1 = [( r mr+ a) { rh + [ rr ( + a) + amr ] } + ahm][ rr ( + a) + am] 33

34 Similarly the innermost stable orbits can be found to be at 1/ r ms = m{3 + A m [(3 A )(3 + A + )] } 1 1 A where 1/ 3 1/ 3 1/ 3 A1 = 1+ (1 a / m ) [(1 + a / m) + (1 a / m) ] 1/ A = ( 3a / m + A1 ) and the upper/lower sign refers to particles orbiting in the same/counter sense as the rotation of the hole. RS / prograde rmin( a = m) = 9RS / retrograde 34

35 The last stable orbit can be shown to correspond to a maximum efficiency of energy extraction which is related to the quantity E, a m = m 4 (1 E ) E 3 3(1 E ). The quantity E decreases from (8/9) 1/ (a=0) to (1/3) 1/ (a=m) for direct orbits, while it increases from (8/9) 1/ to (5/7) 1/ for retrograde orbits. The maximum binding energy 1 E ranges from 1 (8/9) 1/, or ~6% for a Schwarzschile black hole (a = 0), to1 (1/3) 1/ or ~4% for a maximally rotting black hole (a = m). 35

36 The Blandford-Znajek model (i) Basic idea Large-scale magnetic fields tied to the disk extract rotational energy from the disk which is re-supplied by the infall of material in the disk. (The rotational energy of the accreting black hole can also be extracted by electromagnetic effects.) 36

37 As a BH accretes, it would inevitably trap some net magnetic flux due to accreted field lines with connections to infinity. There would then be an approximately time-steady magnetic field configuration with field lines embedded in the hole s event horizon, even while their far ends close at very large distance from the black hole. 37

38 If there are sufficient charge distributions around the black hole to provide the force free condition, the magnetic field lines exert no force and corotate rigidly with the rotating black hole. These rotating field lines induce an electromagnetic force that accelerates charged plasma at relativistic speeds along the axis of rotation, then drive an MHD outflow/jet. A current flow along the jet generates a toroidal field, the magnetic pressure of which, increases away from the axis and so confines the current flow. So a jet is therefore self-collimating. 38

39 39

40 (ii) The force-free condition The motion of an ionized gas with overall charge neutrality is governed by the force density r r r 1 r r f = ρg + qe + j B c r r Note that j = qv is rarely satisfied in MHD since the currents arise from the net motion of positive and negative carriers in a neutral medium, rather than from a net space charge moving with the fluid. Overall neutrality can be maintained if there are regions of opposite net charge. 40

41 There are two limiting cases. (1) If the particle density is sufficiently large the particle inertia will be important and the region above the disk will not be force free. Strong magnetic fields balance gravity. This is the magnetostatic case, which may be important in a magnetized accretion disk, or above the disk if there is a large amount of plasma there. In this case the particles may be accelerated along fieldlines directly, extracting energy and angular momentum from the disk (Blandford and Payne 198). 41

42 () For a sufficiently tenuous plasma with large electric fields gravitational terms can be neglected. We have the force-free case, in which the charges in the magnetosphere are assumed to be sufficiently mobile to allow them to act a sources of field, i.e., r 1 r r qe + j B = 0 c The force-free condition is appropriate in pulsar magnetosphere and in the Blandford-Znajek model. 4

43 Assume that the magnetic energy density in the disk is small compared with the thermal energy density, and the dense fully ionized disk plasma has a high electrical conductivity, the magnetic field will be frozen into the disk and the field will be swept round with the disk material. Now consider a stationary model where / t= / φ=0. Since B φ is independent of φ, we have B v φ = 0. Combined with B v = 0 this leads to B v = 0 p This means that the lines of the poloidal magnetic field rotate with the disk material to which they are anchored. 43

44 The sufficient condition for stationary fields is degeneracy E v B v = 0. Since E φ = 0, this means that v 1 v v v 1 r v Ep B = Ep + ( Ω r) c c From r r 1 B E p = E = = 0 c t we have r v r [( Ω r ) B ] = 0 + Bp p r = 0. 44

45 or s B p r Ω = 0 This means that Ω does not change along the poloidal fieldlines. Therefore the fieldlines cannot wind up. Particles are forced to move away from the disk. This provides a mechanism for generation of relativistic particles. 45

46 (iii) How particles are extracted from the disk? The process is analogous to that in pulsar models. Inside the disk, since σ, there is an electric field r r r E = 1 v B c and hence a charge density r r r q = E = Ω B / c. This charge density gives rise to an electrical field outside the disk, which is available to pull charges out of the disk. V ~ ER ~ V (M/M )(B/10 4 G) 46

47 47 (iv) BH power The electromagnetic power flowing through the force-free region from the disk to the acceleration region is π π π φ φ 4 ~ 4 4 B B R B E c B E c S p p Ω = = v v v v Assume Ω Ω Ω R c R R c B B p ~ ) ( ~ 1/ φ The disk luminosity ergs ) /10 ( G) /10 ( ~ 10 d ) / ( ~ d ~ Θ Ω M M B R R c R c B R R S L π π

48 References: 1. Frank, J., King, A. & Raine, D. 00, Accretion power in astrophysics (CUP). Blandford, R. D. & Znajek, R. L. Electromagnetic extraction of energy from Kerr black holes MNRAS 179:433 (1977) 3. Spruit, H. Theory of magnetically powered jets, arxiv: Krolik, Magnetic extraction of spin energy from a black hole astro-ph/ (000) 48