High-Energy Astrophysics Lecture 6: Black holes in galaxies and the fundamentals of accretion. Overview

Transcription

1 High-Energy Astrophysics Lecture 6: Black holes in galaxies and the fundamentals of accretion Robert Laing Overview Evidence for black holes in galaxies and techniques for estimating their mass Simple physics of black holes Fundamentals of accretion: Energy available Limits to accretion: the Eddington luminosity Thin disks General accretion flows Black hole masses Individual stellar velocities Milky Way (3 x 10 6 solar masses within 001 pc) Water masers in NGC 4258: 36 x 10 7 solar masses within 01 pc (VLBA) Resolved gas kinematics eg M87: 24 x 10 9 solar masses within 18 pc (HST) Stellar velocity dispersion Reverberation mapping Broad Fe Kα line (later) Black hole - bulge luminosity correlation: M BH / M bulge 10-3 Animation of stellar motions in the Galactic centre Orbital motion around the Galactic centre Mass distribution near the Galactic centre 1

2 The Black Hole at the Galactic Centre Water masers in NGC 4258 Galactic centre has a non-thermal source of precisely-known position Observe stellar orbits directly (most recently with adaptive optics) in the near-infrared) Most stringent constraints from star S2 (closest approach): a = 55 light days; period = 152 years; highly elliptical orbit Best model: 26 x 10 6 solar mass point source + star cluster S2 approaches to within 2100R S (see later) for a 26 x 10 6 solar mass black hole NGC 4258 Gas kinematics in M87 Water masers are point tracers of mass They emit at 135cm and can be observed with VLBI (angular resolution 200µas; spectral resolution 02 kms -1 ) Masers in nearly edge-on disk Keplerian rotation, so v = (GM/r) 1/2 M 35 x 10 7 solar masses High precision of Keplerian rotation => point mass Gas disk kinematics in M87 Black hole - bulge mass relation Same principle as masers, but with poorer spatial resolution Spatially-resolved HST spectroscopy of rotating gas disk, emitting optically in the Hα line Again, observe Keplerian rotation Infer central mass 32 x 10 9 solar masses Compare our Galaxy and NGC4258: M87 s black hole is much heavier - but M87 is a much larger elliptical galaxy 2

3 What is a black hole? A black hole is a gravitational singularity, from which electromagnetic radiation cannot escape Simple Newtonian calculation (Michell 1783): escape velocity from a suitably compact star exceeds c, and therefore light cannot escape v = (2GM/r) 1/2 = c r = 2GM/c 2 This is the correct (General relativistic) expression for the Schwarschild radius of a black hole of mass M Define the gravitational radius r g = GM/c 2 Schwarschild black holes Non-rotating black holes are described by the Schwarschild metric r s = 2r g = 2GM/c 2 = 3 km (M/M SUN ) is the Schwarschild radius Event horion For r < r s, there is no photon trajectory which allows escape Last stable orbit occurs at r = 3 r s = 6r g Efficiency of energy release from accretion onto a Schwarschild black hole is related to the binding energy of the last stable orbit The maximum efficiency is 1-8 1/2 = 0057 Kerr black holes Kerr metric describes all rotating black holes They are characterised by the mass M and angular momentum J = amc (0 a 1) only Dragging of inertial frames If a 0 then there are no stationary observers: every physically realisable reference frame must rotate Last stable orbit More complicated forms Radii are different for prograde and retrograde orbits (minimum GM/c 2 for a = 1) Efficiency of energy extraction is higher than for non-rotating holes because the last stable orbit is closer in Maximum value = 1-3 1/2 = 042 Photon propagation near black holes Special relativistic Doppler boosting Gravitational redshift If dt is the proper time interval seen by a distant observer and dt is that seen by an observer close to the black hole, then dt = (1-r s /r) 1/2 dt As r -> r s, events which take a finite amount of time as measured near the black hole appear to take divergently long times when observed at large distances (and radiation is redshifted) Curvature in photon trajectories -> emission line skewed to higher energies X-ray iron lines Image of Fe line emission 3

4 Predicted Fe line profile Observed Fe line profile Evidence for a spinning black hole? Direct evidence for an event horion? In both galactic (see later lecture) and extragalactic sources, evidence from gravitational tracers is for a large amount of mass within a small radius But this does not inevitably require a black hole Argue that there are no stable, massive objects with small enough radii other than black holes (neutron star mass limit; supermassive stars; star clusters, etc) More directly, look for signatures of impact on the surface of an accreting object: thermonuclear bursts in accreting neutron stars, but not black holes Directly image the event horion - X-ray interferometry? The Eddington limit Eddington limit Central source radiates, therefore exerting an outward force on the accreting gas Assuming Thomson opacity only, this sets a maximum luminosity L Edd for the central source, above which radiation overpowers gravity For pure hydrogen plasma: Inward gravitational force = GM(m p +m e )/r 2 GMm p /r 2 Radiation pressure acts on electrons; communicated electrostatically to protons Each photon loses momentum p = hν/c; multiply by photon flux N and cross section σ T to get net radiation force The Eddington limit N = L/4πr 2 hν Hence balance forces: σ T L / 4πr 2 c = GMm p /r 2 Hence limiting luminosity L = L Edd = 4πGMm p /σ T = 13 x (M/M sun ) W For plasma with other elements, replace m p with mass per electron Ways of evading the Eddington limit: non-spherical geometry (not large factors); non-steady-state (eg supernovae) The Eddington limit - related quantities Eddington accretion rate Given an efficiency η, the accretion rate for Eddington luminosity is L edd / c 2 η = 3M 8 (η/01) -1 M sun / year Implied black hole mass for Eddington luminosity: AGN: W => M sun 4

5 Accretion disks Angular momentum is difficult to lose for infalling material Orbit of minimum energy at constant angular momentum is circular - hence disk Viscosity causes loss of angular momentum, so disk material gradually sinks towards the central object, dissipating energy which can potentially be radiated away What is the viscosity? Turbulence Magnetic fields Thin disks Standard model, well established for accreting binary stars has geometrically thin, optically thick disk (alias Shakura-Sunyayev; α-disk α- prescription: ν = α c s H, where ν is the kinematic viscosity, c s is the sound speed, H is the disk scale height and α 1is assumed to be constant Temperature If the emission is black-body and comes from close to the last stable orbit, then T 10 6 L -1/4 39 (L/L Edd ) 1/2 K where L 39 is the luminosity in units of W Hence characteristic temperatures in UV for AGN; X-ray for binary stars Radiatively inefficient accretion Observed accretion with L << L Edd One possibility is just the standard thin-disk solution with a low accretion rate There is another class of accretion flows in which the accretion rate is very low These are: Optically thin Geometrically thick Radiatively inefficient One example: cooling times are very long in tenuous plasma, so material falls into the black hole before it has time to radiate Electromagnetic energy extraction Basic idea Large-scale magnetic fields anchored in the disk extract rotational energy Disk re-supplied by fresh infalling material Blandford-Znajek mechanism Field lines are also anchored on the black hole, allowing its rotational energy to be tapped The thin accretion disk in more detail Equation of hydrostatic equilibrium perpendicular to the disk g is the gravitational potential at radius R and height Assume perfect gas, sound speed c s (independent of height) Integrate to get the density as a function of where 5

6 Thin disks must be supersonic Keplerian rotation Scale height as a function of radius and rotational velocity Therefore, h << R => v rot >> c s, so a thin disk requires supersonic rotation Viscosity Disk material will not accrete onto the central object unless it loses angular momentum This requires some viscosity to transport angular momentum outwards, allowing the material to fall inwards Circular rotating disk, thickness t, viscosity η, angular velocity Ω Tangential force per unit area exerted by disk interior to r on disk exterior to r Force acts over area 2πrt hence torque Γ Viscosity and accretion rate Keplerian rotation in outer part of disk (where angular momentum loss is small Change of angular momentum for inner disk Must equal the change of angular momentum due to inflow of disk material, hence: What is the viscosity mechanism? Reynolds number, where V is the flow speed, L is a typical length scale and ν = η/ρ is the kinematic viscosity Low R => viscous flow; high R => turbulence From kinetic theory (λ = mean free path) => R Flow is turbulent Therefore, kinetic viscosity is irrelevant Turbulence and magnetic fields provide an effective viscosity, but are difficult to calculate Hence the empirical ansat of Shakura & Sunyayev: Turbulent viscosity ν = αc s H, where H = scale height Energy loss rate See Longair, vol 2, 1633 for derivation -de/dt = (3GmM/4πr 3 )[1 - (r*/r) 1/2 ] (per unit surface area of the disk, integrated over height) Energy loss rate is independent of viscosity (which is why we have been able to make progress despite lack of knowledge of the viscosity prescription) To get disc luminosity, integrate -de/dt over area from r* to infinity: L = GmM/2r*, ie half of the total potential energy Slightly different expressions for a black hole (Longair 1634) Temperature Assume disk is optically thick, and that there is sufficient scattering that the emission can be approximated as black-body Equate heat dissipated between r and r + r to 2σT 4 x 2πr r Hence T = (3GmM/8πr 3 σ) 1/4 T varies with radius, so we need to integrate over r to get the overall spectrum of the disk 6

7 Spectrum Integrated spectrum (Longair 1635) ν 2 at low frequencies (Rayleigh-Jeans) ν 1/3 at frequencies corresponding to temperatures of material in the disk exp(-hν/kt) at frequencies above kt in /h 7