High-Energy Astrophysics Lecture 6: Black Holes in Galaxies and The

Home , Schwarzschild radius

High-Energy Astrophysics Overview Lecture 6: Black holes in galaxies and the fundamentals of accretion z Evidence for black holes in galaxies and techniques for estimating their mass. Robert Laing z Simple physics of black holes

z Fundamentals of accretion: Energy available Limits to accretion: the Eddington luminosity Thin disks General accretion flows

Animation of stellar motions in the Black hole masses Galactic centre 6 z Individual stellar velocities Milky Way (3 x 10 solar masses within 0.01 pc)

7 z Water masers in NGC 4258: 3.6 x 10 solar masses within 0.1 pc (VLBA). 9 z Resolved gas kinematics e.g. M87: 2.4 x 10 solar masses within 18 pc (HST).

z Stellar velocity dispersion

z Reverberation mapping

z Broad Fe Kα line (later) Black hole - bulge luminosity correlation: ∼ -3 MBH / Mbulge 10

Orbital motion around the Galactic centre Mass distribution near the Galactic centre

1 The Black Hole at the Galactic Centre Water masers in NGC 4258 z Galactic centre has a non-thermal source of precisely-known position. z Observe stellar orbits directly (most recently with adaptive optics) in the near-infrared). z Most stringent constraints from star S2 (closest approach): a = 5.5 light days; period = 15.2 years; highly elliptical orbit.

6 z Best model: 2.6 x 10 solar mass point source + star cluster. z S2 approaches to within 2100RS (see later) for a 2.6 x 106 solar mass black hole.

NGC 4258 Gas kinematics in M87 z Water masers are point tracers of mass. They emit at 1.35cm and can be observed with VLBI (angular resolution 200µas; spectral resolution 0.2 kms-1). z Masers in nearly edge-on disk. Keplerian rotation, so ∆v = (GM/r)1/2

7 z M ≈ 3.5 x 10 solar masses z High precision of Keplerian rotation => point mass.

Gas disk kinematics in M87 Black hole - bulge mass relation z Same principle as masers, but with poorer spatial resolution. z Spatially-resolved HST spectroscopy of rotating gas disk, emitting optically in the Hα line. z Again, observe Keplerian rotation.

9 z Infer central mass ≈ 3.2 x 10 solar masses. z Compare our Galaxy and NGC4258: M87’s black hole is much heavier - but M87 is a much larger elliptical galaxy.

2 What is a black hole? Schwarzschild black holes

z z A black hole is a gravitational singularity, from Non-rotating black holes are described by the which electromagnetic radiation cannot escape. Schwarzschild metric 2 z Simple Newtonian calculation (Michell 1783): rs = 2rg = 2GM/c = 3 km (M/MSUN) escape velocity from a suitably compact star is the Schwarzschild radius.. exceeds c, and therefore light cannot escape. z Event horizon For r < rs, there is no photon v = (2GM/r)1/2 = c trajectory which allows escape. 2 r = 2GM/c z Last stable orbit occurs at r = 3 rs = 6rg. z This is the correct (General relativistic) expression z Efficiency of energy release from accretion onto a for the Schwarzschild radius of a black hole of Schwarzschild black hole is related to the binding mass M. energy of the last stable orbit. The maximum 2 efficiency is 1 - 81/2 = 0.057. z Define the gravitational radius rg = GM/c

Kerr black holes Photon propagation near black holes z Kerr metric describes all rotating black holes. They z Special relativistic Doppler boosting are characterised by the mass M and angular z Gravitational redshift If dt is the proper time momentum J = aMc (0 ≤ a ≤ 1) only. interval seen by a distant observer and dt’ is that z Dragging of inertial frames If a ≠ 0 then there are seen by an observer close to the black hole, then no stationary observers: every physically realisable dt’ = (1-r /r)1/2 dt reference frame must rotate. s As r -> rs, events which take a finite amount of time z Last stable orbit More complicated forms. Radii as measured near the black hole appear to take are different for prograde and retrograde orbits divergently long times when observed at large 2 (minimum GM/c for a = 1). distances (and radiation is redshifted). z Efficiency of energy extraction is higher than for z Curvature in photon trajectories non-rotating holes because the last stable orbit is closer in. Maximum value = 1 - 31/2 = 0.42. -> emission line skewed to higher energies.

X-ray iron lines Image of Fe line emission

3 Predicted Fe line profile Observed Fe line profile

Evidence for a spinning black hole?

Direct evidence for an event horizon? The Eddington limit z In both galactic (see later lecture) and extragalactic z Eddington limit Central source radiates, therefore sources, evidence from gravitational tracers is for a exerting an outward force on the accreting gas. large amount of mass within a small radius. But this Assuming Thomson opacity only, this sets a does not inevitably require a black hole. maximum luminosity LEdd for the central source, z Argue that there are no stable, massive objects above which radiation overpowers gravity. For pure with small enough radii other than black holes hydrogen plasma: (neutron star mass limit; supermassive stars; star Inward gravitational force clusters, etc.) 2 ≈ 2 = GM(mp+me)/r GMmp/r z More directly, look for signatures of impact on the surface of an accreting object: thermonuclear Radiation pressure acts on electrons; bursts in accreting neutron stars, but not black communicated electrostatically to protons. Each holes. photon loses momentum p = hν/c; multiply by σ photon flux N and cross section T to get net z Directly image the event horizon - X-ray radiation force. interferometry?

The Eddington limit The Eddington limit - related quantities

2 N = L/4πr hν z Eddington accretion rate Given an efficiency η, the accretion rate for Eddington luminosity is Hence balance forces: 2 η η -1 σ π 2 2 Ledd / c = 3M8( /0.1) Msun / year T L / 4 r c = GMmp /r z Implied black hole mass for Eddington luminosity: z Hence limiting luminosity 36 40 5 8 π σ 31 AGN: 10 - 10 W => 10 - 10 Msun L = LEdd = 4 GMmp/ T = 1.3 x 10 (M/Msun) W z For plasma with other elements, replace mp with mass per electron. z Ways of evading the Eddington limit: non-spherical geometry (not large factors); non-steady-state (e.g. supernovae).

4 Accretion disks

z Angular momentum is difficult to lose for infalling material. Orbit of minimum energy at constant angular momentum is circular - hence disk.

z Viscosity causes loss of angular momentum, so disk material gradually sinks towards the central object, dissipating energy which can potentially be radiated away.

z What is the viscosity? Turbulence Magnetic fields

Thin disks Radiatively inefficient accretion z Standard model, well established for accreting z Observed accretion with L << LEdd binary stars has geometrically thin, optically thick z One possibility is just the standard thin-disk solution disk (alias Shakura-Sunyayev; α-disk. with a low accretion rate. z α- prescription: ν = α c H, where ν is the kinematic s z There is another class of accretion flows in which viscosity, c is the sound speed, H is the disk scale s the accretion rate is very low. These are: height and α ≤ 1is assumed to be constant. Optically thin z Temperature If the emission is black-body and comes from close to the last stable orbit, then Geometrically thick ∼ 6 -1/4 1/2 Radiatively inefficient. T 10 L39 (L/LEdd) K 39 z One example: cooling times are very long in where L39 is the luminosity in units of 10 W. Hence characteristic temperatures in UV for AGN; tenuous plasma, so material falls into the black hole X-ray for binary stars. before it has time to radiate.

Electromagnetic energy extraction The thin accretion disk in more detail z Basic idea Large-scale magnetic fields anchored in Equation of hydrostatic equilibrium the disk extract rotational energy. perpendicular to the disk. z Disk re-supplied by fresh infalling material. gz is the gravitational potential at radius R z Blandford-Znajek mechanism Field lines are also and height z. anchored on the black hole, allowing its rotational energy to be tapped. Assume perfect gas, sound speed cs (independent of height).

Integrate to get the density as a function of z where

5 Thin disks must be supersonic Viscosity

Keplerian rotation 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 Scale height as a function of radius and material to fall inwards. rotational velocity. Circular rotating disk, thickness t, viscosity η, angular velocity Ω. Therefore, h << R => vrot >> cs, so a thin disk requires supersonic rotation. 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 What is the viscosity mechanism?

Keplerian rotation in outer part Reynolds number, where V is the flow speed, of disk (where angular momentum L is a typical length scale and ν = η/ρ is the loss is small. kinematic viscosity. Low R => viscous flow; high R => turbulence. Change of angular momentum for inner disk From kinetic theory (λ = mean free path) => R ∼ 1012. Must equal the change of angular momentum due to inflow Flow is turbulent. Therefore, kinetic viscosity is irrelevant. of disk material, hence: Turbulence and magnetic fields provide an effective viscosity, but are difficult to calculate. Hence the empirical ansatz of Shakura & Sunyayev:

ν α Turbulent viscosity = csH , where H = scale height

Energy loss rate Temperature

z See Longair, vol 2, 16.3.3 for derivation. z Assume disk is optically thick, and that there is

. 3 1/2 sufficient scattering. that the emission can be z -dE/dt = (3GmM/4πr )[1 - (r*/r) ] (per unit surface approximated as black-body. area of the disk, integrated over height). 4 z Equate heat dissipated between r and r + ∆r to 2σT z Energy loss rate. is independent of viscosity (which π ∆ is why we have been able to make progress despite x 2 r r . 3 1/4 lack of knowledge of the viscosity prescription). z Hence T = (3GmM/8πr σ)

z To get disc luminosity, integrate -dE/dt over area z T varies with radius, so we need to integrate over r from r* to infinity: to get the. overall spectrum of the disk L = GmM/2r*,. i.e. half of the total potential energy.

z Slightly different expressions for a black hole (Longair 16.3.4)

6 Spectrum z Integrated spectrum (Longair 16.3.5) ν2 at low frequencies. (Rayleigh-Jeans) ν1/3 at frequencies corresponding to temperatures of material in the. disk. ν exp(-h /kT) at frequencies above kTin/h.