Compact Objects and Relativistic Astrophysics Course

Image Credit: NASA

DARK Cosmology Centre - Niels Bohr Institute - May 23, 2011

Martin PessahNiels Bohr International [email protected]

Compact Objects and Relativistic Astrophysics CourseLecture on ‘Accretion Disks’

mailto:[email protected]

mailto:[email protected]

Why Are Accretion Disks Important?

Proto-star X-ray Binary Active Galactic Nucleus

Crucial for understanding- How stars and planets form- What powers the brightest X-ray sources in the sky- Why Active Galactic Nuclei (Quasars) shine- How to use flow dynamics to map black hole space-time

Release of gravitational energy in accretion disks responsible for some of the most powerful phenomena in nature!

A Natural Outcome

* To a first approximation the gas is falling into a central potential

* Angular momentum is mostly conserved

* Gas can cool down faster than it can get rid of angular momentum

* Flattened, rotating structure (AKA disk!) forms...

Imagine a cloud of gas collapsing due to its own gravity

Typical Masses, Sizes, and Luminosities

From ‘Black Hole Accretion Disks’, Kato, Fukue & Mineshige, 1998

*Observational Evidence

* Basics of Accretion Disk Physics

* Modern Accretion Disk Theory & MHD Turbulence

* Numerical Simulations

Overview

Observational Evidence

Protoplanetary Disks

From ‘Black Hole Accretion Disks’, Kato, Fukue & Mineshige, 1998

Remillard & McClintock, 2006

X-ray Binary Disks

AGN Disks

Evidence for Disks

From ‘Accretion Power in Astrophysics’; Frank, King, & Raine, 1995

Evidence for Disks


Evidence for Disks


Black Hole States

Remillard & McClintock, 2006

Meyer-Hofmeister et al. 2009

Extreme changes in luminosity, spectra, &

variability !!!

Variability at All Mass Scales!

Axelsson et al. 2006 McHardy et al. 2007

Relativistic Iron Lines

Broad iron lines in AGN (Fabian et al.)Sensitive to ‘inner edge’

Sensitive to inclination

Detailed modeling of lineprofile allows us to ‘map’

the space-time aroundblack holes

Basics of Accretion Disk Physics

Why Are Accretion Disk so Hard to Understand?

Magnetic fields do not seem to influence stellar structure significantly

Gravity balanced bypressure gradient along REnergy flows along R too!

R

Magnetic fields are essential for accretion

disks to work

Rφz

MassMomentumEnergy

Non-thermal processesMostly thermal energy

R

vK =

GM

R

ΩK =

GM

R3

Some Dynamical Considerations

If the disk were a collection of non-interacting particles there would be no accretion

Particles in a central potential move in stable Keplerian orbits

Keplerian Disks 101

Ω(R)

R

Ω ∼ R−3/2

Ω2 =GM

R3

Matter

R

l(R)

l = vφR = (ΩR)R = ΩR2

l ∼ R1/2

Importance of Angular Momentum Transport

Gas in the disk must lose angular momentum!!!!

Angular Momentum

∂ρ

∂t+∇.(ρv) = 0

Importance of Stress...

∂l

∂t+∇.(lv) = −1

r

∂

∂r(r2Trφ)

If there is no stress, angular momentum for fluid elements in the disk is conserved and matter does not accrete!

Mass conservation

∂l

∂t+∇.(lv) = 0 Angular momentum

is not conserved....

Solving the Angular Momentum Problem. I

Problem I: Transport of Angular Momentum

!!!

Re~10Re~13Re~26Re~140Re~2000Re~10000

Guadalupe Island vortex street movie from GOES

Re =V L

ν

Courtesy of CK Chan

Reynolds numbers in accretion disks are HUGE!!!

Re =V L

ν∼ 1010 − 1016

Solving the Angular Momentum Problem. I

Solution I: Assume some kind of “turbulent” viscosityEddies of size ‘H’ interacting with turnover velocity ‘α cs’

Shakura & Sunyaev ‘70s, α-model

tturb ∼ tobs !!!

With this enhanced stresswe can match the fast timescales observed!

The Standard Accretion Disk Model

Σ, vr, vφ, P, H, cs, Teff , Tc, τ, κ

The Standard Accretion Disk Model

T 4eff = T 4

c /3τ

τ = κΣ/2κ = κ(Σ, Tc)

cs =

kBTc/µmH

Wide Range of Physical Conditions


Structure of Standard Accretion Disk Model

We can now calculate the emitted spectrum...

Spectral Energy Distribution

From ‘The Physics of Compact Objects’; Shapiro & Teukolsky’; 1983

Assume energy generated by stresses responsible for angular momentum transport is locally radiated away.

Disk can be divided in regions

1. Outer: P_gas and free-free

2. Middle: P_gas and e scattering

3. Inner: P_rad and e scattering

Summary of Standard Accretion Disk Model

Since the early 70’s we have used this ‘viscosity’ prescription to remove angular momentum from the disk...

Matter

Angular momentum

Σ(r), P (r), T (r), vr(r), ...We can calculate the global structure of the disk

Modern Accretion Disk Theory&

MHD Turbulence

Solving the Angular Momentum Problem. II

Problem II: Origin of Turbulence? - Keplerian flows are very STABLE to hydrodynamic perturbations (Hawley et al. 1999, Yi et al. 2006) - Convection? (Stone & Balbus 1996, Lesur & Ogilvie 2010)

Solution II: Magnetic Fields* Velikhov & Chandrasekhar (early 60’s) Balbus & Hawley (early 90’s)

* Mechanism to disrupt laminar flows

* Numerical simulations confirm development of MHD turbulence

B

Turbulent Magnetized Accretion Disks: MRI

We need to understand the dynamics of magnetic fields in differentially rotating plasmas!!!

Differentially rotating magnetized plasmas are unstable to the Magnetorotational Instability (MRI)

(Velikhov & Chandrasekhar, 60’s; Balbus & Hawley, 90’s)

From J. Hawley’s websiteFrom J. Stone’s website

A Piece of Physics:The Magnetorotational Instability

P

Restoring force: pressure gradients

δP

k

Acoustic Waves

B

Restoring force: magnetic tension

δB

k

Alfven Waves

φ

r

MHD Waves

BzMagnetic Field

r

z

φ

r

zBzδBφ

φ

Magnetic Field

Magnetic Tension

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

φ

r

r

zBzδBφ

φ

Magnetic Field

MHD Waves

φ

r

r

zBzδBφ

φ

Magnetic Field

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ k

φ

Magnetic Field

φ

r

MHD Waves

r

zBzδBφ k

φ

Magnetic Field

€

vA2 =

B2

4πρ€

ω 2 = k 2vA2

φ

r

MHD Waves

Magnetorotational Instability

To the black hole

BzMagnetic Field

r

z

φ

Differential rotation

φ

r

r

zBzδBφ


φ

Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

φ

r

z


Magnetic Field


φ

r

r

z

BzδBφ


MagneticTorques

φ

Magnetic Field



+

φ

rBzδBr

r

z

BzδBφ


MagneticTorques

φ

Magnetic Field



+

BzδBr

€

vA2 =

B2

4πρ€

k 2vAz2 < −2Ω2 d lnΩ

d ln r

MHD Equations

Local -shearing box- MHD Equations

Use background field and rotation rate to define characteristic scales in the problem

Linear Mode Analysis

A→ A + δADynamical equations = 0

Dynamical equations for δ = 0

p(k, ω) = 0Obtain the dispersion relation

δρ δvNeglect 2nd order terms

Expand in Fourier Series

δA ∼ δρ, δP, δv, δB →

Ak exp[−i(k.x + ωt)]

63

ω = ω(k)Find the eigenfrequencies

Growth Rates with Dissipation

z

r

k B

Various limits studied by Sano et al. 1998, Lesaffre & Balbus 2007, Lesur & Longaretti 2007, and many others…

Pessah & Chan, 2008

Angular Momentum Transport: MRI & Turbulence

Sano et al. 2004

Maxwell Stress

Reynolds Stress

MRI and MHD turbulence lead to correlated fluctuations

of the PROPER sign!!!

€

∂l∂t

+∇ ⋅ (lv) = −1r∂∂r[r2(Rrφ −Mrφ )]

Angular momentum conservation

Saturation of MRI & MHD Turbulence

-Maxwell

Reynolds

MRI growth

SaturationNon-linear

Insets from Sano et al. 2004

Numerical Simulations

Importance & Limitations of Numerical Simulations

1Gyr

1yr

1sec

0.1AU 1AU 1000AU

Mean Field Models

BroadLine

Region

Inside Horizon

NumericalSimulations

Viscous

Orbital Saturation

BHGrowth

BHVariability

Hubble Time

107Mo

Disk Spectrum

MHD Turbulent Transport vs. alpha-viscosity

Pessah, Chan, & Psaltis, 2006Pessah, Chan, & Psaltis, 2008Shakura & Sunyaev, 1973

Standard prescription does not capture

physics correctly!

Need to understandturbulent MHD

angular momentum transport from first

principles...

Shakura & Sunyaev (‘70s):transport due to turbulence

Current Focus and Future Prospects

Davis, Stone, & Pessah 2010

CK Chan & Pessah 2011

Better understanding of micro-physics. Thermodyn.

Synergy between analytical and numerical work

More realistic global simulations

Global 3D MHD Simulations

Beckwith et al. 2011

Global 3D MHD Simulations

Beckwith et al. 2011

α−stress (~ P)Chan, Psaltis, & Ozel, ‘06

Global Models for Accretion onto Black Holes

Risco

RSch

Simulations showstresses DO NOTvanish inside ISCO

It is usually argued that stresses vanish

at the inner disk

Implications: * Inner disk structure * Spin measurements * Emission inside ISCO * Spin evolution

Accretion Disks Big Picture Recap

Since early 70’s we have dealt with this problem by using a‘viscosity’ prescription proposed by Shakura & Sunyaev (1973)

Fundamental Problem in Accretion Disks: How to get rid of angular momentum…

Balbus & Hawley (1991): magnetic fields are key!Differentially rotating, magnetized plasmas are unstable to MRI

Ensuing turbulence removes angular momentum from disk!

Massive analytical and numerical efforts

to understand MHD turbulence and

simulate accretion disksFrom J. Hawley’s websiteFrom J. Stone’s website

Bibliography:

Black holes in binary systems. Observational appearance.Shakura, N. I.; Sunyaev, R. A. 1973A&A....24..337S

Theory Of Accretion Disks I: Angular Momentum Transport Processes Papaloizou, J. C. B.; Lin, D. N. C. 1995ARA&A..33..505P

Theory of Accretion Disks II: Application to Observed SystemsLin, D. N. C.; Papaloizou, J. C. B. 1996ARA&A..34..703L

Balbus, Steven A.; Hawley, John F. Turbulent transport in accretion disks1998AIPC..431...79B

Accretion disc viscosity: how big is alpha?King, A. R.; Pringle, J. E.; Livio, M. 2007MNRAS.376.1740K

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Compact Objects and Relativistic Astrophysics Course