study material “accretion power in astrophysics”, frank
TRANSCRIPT
(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