(HYPER)-ACCRETIONMaster course COA

April 2018

E.M. Rossi, Leiden Observatory

Study material “Accretion power in astrophysics”, Frank, King, Raine

EDDINGTON LIMIT A luminosity limit

Point mass that radiates isotropically

F =L

4⇡r2

photon energy E has momentum E/c

momentum flux isF

c=

L

4⇡cr2force per unit area

frad

pe-

frad =L�T

4⇡cmpr2=

LT

4⇡r2force per unit mass

Thompson cross section

Thompson opacityT =�T

mp

mp ⇡ 2000 me

(section 1.2)

c

EDDINGTON LIMIT A luminosity limit

frad

pe-

Equality of radiation pressure on the completely ionised matter and of the gravitational forces of attraction to the BH

fgrav

fgrav =GM

r2frad =

LT

4⇡r2=

LEdd =4⇡GM

T⇡ 1038

✓M

M�

◆erg/s

c

c

EDDINGTON LIMIT : MASS ACCRETION RATE

Formally there is NOT an accretion rate limit (Begelman 78,79, e.g. Alexander & Natarajan 2014)

Mcr = 3⇥ 10�8

✓0.06

⌘

◆✓M

M�

◆M� yr�1

Mcr = LEdd/(⌘c2) ⌘ ' 0.06� 0.4

let’s discuss it…

TRAPPING RADIUS

the trapping radius is radius within which the local optical depth τ (r) ∼ κρ(r)r makes photon diffusion outward slower than accretion inward

tdi↵ = tdyn Rtr

c/⌧=

Rtr

v⌧ =

c

v⇡ ⇢(r)r � 1

Rtr ⇡ Rs

M

Mcr

!

M = 4⇡⇢vR2t Mcr = LEdd/(⌘c

2) Rs =2GM

c2

M � Mcrso it can be that because radiation is advected inward

Begelman 78,79

tr

G. THIN-O. THICK ACCRETION DISC

• Thin —> H << R, where H is disc height (density scale for T=const)

• Hydrostatic equilibrium in z-direction

• Orbits are circular and tangental velocity nearly Keplerian

viscous or drift timescale

kinematic viscosity

• Accretion due to shear viscosity between orbits

r>>Rin vr ' � R

tvisc(R) tvisc(R) ' R2

⌫

v� 'r

GM

Rtorb =

2⇡R

v�=

2⇡

⌦

vz = 0

RR

⌦ = Rv�

G. THIN-O. THICK ACCRETION DISC

• Thin —> H << R, where H is disc height (density scale for T=const)

• Hydrostatic equilibrium in z-direction

• Orbits are circular and tangental velocity nearly Keplerian

• Accretion due to shear viscosity between orbits

v� 'r

GM

Rtorb =

2⇡R

v�=

2⇡

⌦

vz = 0

VISCOSITY TO DRIVE ACCRETIONStudy material “Accretion power in astrophysics”, sections 4.6, 4.7

“Accretion discs can be an efficient machine for slowly lowering material in the gravitational potential of an accreting object and extracting the energy as radiation”

let’s first consider plane differential rotation

chaotic motion causes transport of momentum orthogonal to the gas motion: this transport process is called “shear viscosity”.”

typical scale and velocity of chaotic motion

how?

VISCOSITY TO DRIVE ACCRETIONStudy material “Accretion power in astrophysics”

“Accretion discs can be an efficient machine for slowly lowering material in the gravitational potential of an accreting object and extracting the energy as radiation”

typical scale and velocity of chaotic motion

�xz = ⌘@u

@z⇡ �⇢v�

@u

@z

x-component of the force per unit area through a surface of constant z

let’s first consider plane differential rotation

VISCOSITY TO DRIVE ACCRETIONStudy material “Accretion power in astrophysics”

“Accretion discs can be an efficient machine for slowly lowering material in the gravitational potential of an accreting object and extracting the energy as radiation”

�xz = ⌘@u

@z⇡ �⇢v�

@u

@z

x-component of the force per unit area through a surface of constant z

⌫ = ⌘/⇢ = fv�

kinematic viscosity

let’s first consider plane differential rotation

VISCOSITY TO DRIVE ACCRETIONStudy material “Accretion power in astrophysics”

“Accretion discs can be an efficient machine for slowly lowering material in the gravitational potential of an accreting object and extracting the energy as radiation”

the Keplerian rotation law implies differential rotation v� 'r

GM

R

\phi-component of the force per unit area through a surface of constant R

kinematic viscosity

the Keplerian rotation law implies differential rotation

�R,� = �⌘Rd⌦

dR= �⌫

⇢Rd⌦

dR

⌫ = ↵Hcssound speed

the viscous force generate a net torque

that makes the ring to lose angular momentum

and flow in, towards low angular mom. radii

VERTICAL STRUCTURE

P ' ⇢c2s@P

@z⇠ P/H

highly supersonic, if the disc is maintained cold

the condition implies

section 5.3

RADIAL STRUCTURE

Euler eq.

v2RR

�v2�R

+c2sR

+v2KR

⇡ 0

⌧ v�since

section 5.3

TEMPERATURE STRUCTURE

Ldisc =GMM

2REnergy to bring matter from infinity, half still in kinetic form

Ldisc ⇠ 2�T 42⇡R2 ! T 4 ⇠ MM

R3

Note: the radiation pressure force scales as the vertical component of gravity

Fg ⇠ MH

R3

L

L

TEMPERATURE STRUCTURE

Ldisc =GMM

2REnergy to bring matter from infinity, half still in kinetic form

Ldisc ⇠ 2�T 42⇡R2 ! T 4 ⇠ MM

R3

Note: the radiation pressure force scales as the vertical component of gravity

Fg ⇠ MH

R3

L

Lacc

TEMPERATURE STRUCTURE

Ldisc =GMM

2REnergy to bring matter from infinity, half still in kinetic formLacc

WD: optical emitter

NS, solar BH: X-ray emitter

Note: no dependence on viscosity!!!

supermassive BH: UV emitter

HYPERACCRETING FLOW PROPERTIES

•Eddington flows are radiation dominated

•Hot: electron scattering dominates

Shakura & Sunyaev 73

1

⇢

dP

dz= �GM

R3z �1

⇢

dPrad

dz=

Fradk

c

L

LEdd=

1

� + 1

� =Pgas

Prad

(assuming constant and z~R)GM

R2⇠ L(1 + �)k

4⇡cR2L ' LEdd

(1 + �)

•Eddington flows are thick, quasi spherical: from hydrostatic equilibrium in vertical direction

H

R⇠ constant in R

H

2Rs⇡ M

Mcrit

! 1, M ! 12Rin

�

P = Pgas + Prad

INFLOW-OUTFLOW MODELSShakura & Sunyaev 73, Blandford & Begelman 99

the luminosity as function of radius

Rsp ⌘ Rtr ⇡ M

M cr

!Rs

L(R) ⇡ GMBHM

R=

GMBHMcr

Rs

✓Rph

R

◆

Wind transports away the matter in excess!

Rsp

R > Rph ! L(R) < LEdd

R < Rph ! L(R) > LEdd

Rsp

Rsp

let’s capped the luminosity to Eddington:

R < Rsp M / R

INFLOW-OUTFLOW MODELSShakura & Sunyaev 73, Blandford & Begelman 99

The total luminosity (integrated over the whole disc is therefore

L ⇡ LEdd

1 + ln

M

Mcr

!

ln(10)=2.3

1+2.3~3

ACCRETION RATE AND BH GROWTH

In the inflow-outflow model the mass accretion at the inner disc radius:

M(Rin) = Mcr

MBH(t) = MBH(0) exp

✓1� ⌘

⌘

t

tEdd

◆

tEdd = 0.45 Gyr

ACCRETION RATE DETERMINES THE PHYSICS

Shakura & Sunyaev 73

Lynden-Bell & Pringle 74

Thorne & Page 74

Pringle 81

1

hyperaccretion

10-2

geometrically thick,

optically thin disc

see ref in this class

geometrically thin,

optically thick disc

Narayan & Yi 94

Abramowicz’s work

Begelman & Blandford

Quataert & Gruzinov

ADAFs, ADIOS, CDAF

radiative inefficient

?

M

Mcr

radiative efficient

“Hyper-accretion:

a possible framework.

THE OUTCOME DEPENDS ON CIRCULARISATION VS TRAPPING

2 classes of models+1

CIRCULARISATION VS TRAPPING

Centrifugal barrier: radius at which the flow circularise and cannot proceed father in its fall

•If

lflow =p

GMRc Rc= circularisation radius

inflow-outflow model just described

i.e.p

GMBHRtr ⌧pGMBHRc Rtr ⌧ Rc

CIRCULARISATION VS TRAPPING

•If

lflow =p

GMRc Rc= circularisation radius

Centrifugal barrier: radius at which the flow circularise and cannot proceed father in its fall

ZEBRA, quasi-star, supra exponential accretion…

(Begelman, EMR, Armitage 08; Volonteri & Rees; Dotan, EMR, Shaviv 11; Alexander&Natarajan 14; Coughlin & Begelman 14;; Begelman & Volonteri 16; Fiacconi& Rossi 17…)

the gas falls and deposits all its energy and angular momentum

well within the trapping radius: outcome still very uncertain

Rtr � Rc

pGMBHRtr �

pGMBHRc

CIRCULARISATION VS TRAPPING

Rtr � Rc

M(Rin) � Mcr

The accretion rate through the inner radius can be arbitrarily large

A way to grow supermassive black holes at high redshift..

HYPER ACCRETING REGIME

1

slim disc

102

(quasi) spherical flow

1011

geometrically

thin neutrino

cooled discsRtr< Rd Rtr>Rd

M

Mcr

“going at even high accretion rate…neutrino cooled discs

e.g. Popham + 99;Lemoine 02; Pruet + 03, Beloborodov 03,10…

NEUTRINO COOLED DISCS

Neutron star mergers and core collapse can generate disc with > 0.01 Msun/s

e.g. collapses timescale = 10s and accreted mass= 1 solar mass

so…these are subcritical disc for neutrinos

⇢ ⇠ 1011g cm�3 T > 1 MeVextreme conditions—->

neutrino productions !

M

extremely supercritical for photons!!M

Mcr

> 1013!

Mcr,⌫ = Mcr

✓�T

�⌫

◆' Mcr ⇥ 1019!!

NEUTRINO COOLED DISCS: PROPERTIES

⇢ ⇠ 1011g cm�3 T > 1 MeV

In the inner part

—neutrino productions

—photon trapped

—electron mildly degenerate

==> neutron rich disc

scrivere beta decay

THE RADIAL STRUCTURE

The neutrino-cooled disk forms above a critical accretion rate M ign that depends on the black hole spin. The disk has an ‘‘ignition’’ radius rign where neutrino flux rises dramatically, cooling becomes efficient, and the proton-to-nucleon ratio Ye drops. Other characteristic radii are r , where most of -particles are disintegrated, r , where the disk becomes -opaque, Beloborodov 2010

wind

inefficient

cooling

neutron-rich

EMR, Beloborodov & Rees 05