astrophysical accretion - nasa · the right mass m within the available time (the growth problem)...
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Astrophysical Accretion
Andrew King
Theoretical Astrophysics Group, University of Leicester, UK
Accretion Power in Astrophysics, Frank, King, Raine, CUP, 2002
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accretion = release of gravitational energy from infalling matter
matter falls infrom distance
energy released as electromagnetic(or other) radiation
accreting object
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If accretor has mass M and radius R, gravitational energy release/mass is
R
GMEacc
this accretion yield increases with compactness M/R: for a given Mthe yield is greatest for the smallest accretor radius R
e.g. for accretion on to a neutron star )10,( kmRMM sun
gmergEacc /1020
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compare with nuclear fusion yield (mainly H He)
gmergcEnuc /106007.0 182
Accretion on to a black hole releases significant fraction of rest—mass energy:
2//2 22 cEcGMR acc (in reality use GR to compute binding energy/mass: typical accretion yield is roughly 10% of rest mass)
This is the most efficient known way of using mass to get energy:
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accretion on to a black hole must power the most luminousphenomena in the universe
Quasars: sergL /1046
McM
R
GMLacc
2
requires yrMM sun /1
X—ray binaries: sergL /1039 yrM sun /10 7
Gamma—ray bursters: sergL /1052 sec/1.0 sunM
NB a gamma—ray burst is (briefly!) as bright as the rest of the universe
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Accretion produces radiation: radiation makes pressure – can thisinhibit further accretion?
Radiation pressure acts on electrons; but electrons and ions (protons)cannot separate because of Coulomb force. Radiation pressure force on an electron is
(in spherical symmetry).Gravitational force on electron—proton pair is
2
)(
r
mmGMF ep
grav
24 cr
LF T
rad
)( ep mm
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thus accretion is inhibited once gravrad FF , i.e. once
sergM
McGMmLL
sunT
pEdd /10
4 38
Eddington limit: similar if no spherical symmetry: luminosity requiresminimum mass
bright quasars must have
brightest X—ray binaries
In practice Eddington limit can be broken by factors ~ few, at most.
sunMM 810
sunMM 10
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Eddington implies limit on growth rate of mass: since
we must have
where
is the Salpeter timescale
T
pacc
c
GMm
c
LM
4
2
teMM 0
yrGm
c
p
T 71054
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Emitted spectrum of an accreting object
Accretion turns gravitational energy into electromagnetic radiation.Two extreme possibilities:
(f) all energy thermalized, radiation emerges as a blackbody.Characteristic temperature , where
i.e. significant fraction of the accretor surface radiates the accretionluminosity. For a neutron star near the Eddington limit
bT
41
24
R
LT acc
b
KTkmRsergL b738 1010,/10
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(b) Gravitational energy of each accreted electron-proton pairturned directly into heat at (shock) temperature . Thus
For a neutron star
Hence typical photon energies must lie between
i.e. we expect accreting neutron stars to be X—ray or gamma—ray sources: similarly stellar—mass black holes
sT
R
GMmkT p
s 3
KTs11105
MeVkThkeVkT sb 501
Good fit to gross properties of X—ray binaries
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For a white dwarf accretor, mass = solar, radius =Find
so UV – X—ray sources.
Gross fit to gross properties of cataclysmic variables (CVs).
Many of these show outbursts of a few days at intervals of a few weeks – dwarf novae. See later….
light
time
cm910
KTKT sb85 10,10~
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cmcGMRMM sun1328 103/2,10
For supermassive black holes we have
so
and is unchanged. So we expect supermassive BH to be ultraviolet, X—ray and possibly gamma—ray emitters.
KKMMT sunb54/17 10)(10
sT
Good fit to gross properties of quasars
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Modelling accreting sources
To model an accreting source we need to
(e) choose nature of compact object – black hole, neutron star, …to agree with observed radiation components
(b) choose minimum mass M of compact object to agree with luminosity via Eddington limit
Then we have two problems:
(13)we must arrange accretion rate to provide observed luminosity, (the feeding problem) and
(2) we must arrange to grow or otherwise create an accretor of the right mass M within the available time (the growth problem)
M
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Examine both problems in the following, for accretingbinaries and active galactic nuclei (AGN)
for binariesfeeding: binary companion stargrowth: accretor results from stellar evolution
for AGNfeeding: galaxy mergers?growth: accretion on to `seed’ black hole?
Both problems better understood for binaries, so develop ideas and theory here first.
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Modelling X—ray binaries
Normal galaxies like Milky Way contain several 100 X—ray point sources with luminosities up to
Typical spectral components ~ 1 keV and 10 – 100 keV
Previous arguments suggest accreting neutron stars and black holesBrightest must be black holes.
Optical identifications: some systems are coincident with luminous hot stars: high mass X—ray binaries (HMXB).
But many do not have such companions: low mass X—ray binaries(LMXB).
serg /1039
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Accreting Black Holes in a Nearby Galaxy (M101)
OPTICAL X-RAY
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Mass transfer in X—ray binaries
Formation: starting from two newly—formed stars in a suitable binary orbit, a long chain of events can in a few rare cases leadto a BH or NS in a fairly close orbit with a main sequence star.
a
1M2M
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Two processes now compete to start mass transfer:
1. Binary loses angular momentum, to gravitational radiation or other processes. Binary separation a shrinks as
— full relation is
where
21
21
M
GaMMJ
2Ja
21 MMM
2. Normal star evolves to become a giant, so radius increases to significant fraction of separation a
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In both cases is continuously reduced.
Combined gravitational—centrifugal (Roche) potential has two minima (`valleys’) at the CM of each star, and a saddle point (`pass’: inner Lagrange point ) between them.
Once sufficiently reduced that the normal star reaches this point, mass flows towards the compact star andis controlled by its gravity – mass transfer
aR2
aR2
1L
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Mass transfer changes the binary separation itself:orbital a.m. J and total binary mass M conserved, sologarithmic differentiation of J implies
and with
we have
a
a
M
M
M
M
J
J
20
2
2
1
1
021
MM
21
212 )(
2 MM
MMM
a
a
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Binary widens if accretor is (roughly) more massive than donor, shrinks if not.In first case mass transfer proceeds on timescale of decrease of , i.e. a.m. loss or nuclear expansion:
or
these processes can drive mass transfer rates up to ~ depending on binary parameters (masses, separation)
aR /2
J
JMM
22
nuct
MM 2
2
yrM sun /10 6
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In this case –stable mass transfer — star remains exactlysame size as critical surface (Roche lobe):
if lobe shrinks relative to star, excess mass transferred veryrapidly (thermal or dynamical timescale)
if lobe expands wrt star, driving mechanism (a.m. loss or nuclearexpansion) rapidly restores contact
Thus binary separation evolves to maintain this equality. Orbitalevolution follows stellar radius evolution.
E.g. in some cases star expands on mass loss, even though a.m. lossdrives evolution. Then orbit expands, and mass transferred toensure that new wider binary has lower angular momentum.
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If instead the donor is (roughly) more massive than accretor, binaryshrinkage mass transfer may increase exponentially on dynamical timescale ~ few orbital periods.
Likely result of this dynamical instability is a binary merger:timescale so short that unobserved.
High mass X—ray binary phase shortlived: accretion actually from wind of hot star. Many binaries pass through HMXB stage. Once donor fills lobe get huge transfer rate: SS433 – thermal timescale mass transfer at ~
Low mass X—ray binaries can have very long lifetimes, ~ a.m.or nuclear timescales
`Paradox’: we observe bright LMXBs in old stellar populations! —see later……
yrM sun /10 4
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Accretion disc formation
Transferred mass does not hit accretor in general, but must orbit it
— initial orbit is a rosette, but self—intersections dissipation energy loss, but no angular momentum loss
Kepler orbit with lowest energy for fixed a.m. is circle.
Thus orbit circularizes with radius such that specific a.m. j is the same as at 1L
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Kepler’s law for binary requires or
, P = orbital period, j = specific a.m.,
Now roughly halfway across binary, and rotates with it, so specific a.m. around accretor of matter leaving it is
So new circular orbit around accretor has radius r such that
, which gives
2
2
a
j
P/2
,/ 22 aaGM 2/13)/( aGM
4
)( 2/1GMa
2/11 )( rGMj
2/1)(GMaj
1L
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816162
1
Raa
M
Mr circ
In general compact accretor radius is far smaller than :
typically donor is at least as large as a main—sequence star, with
A neutron—star accretor has radius
and a black hole has Schwarzschild radius
and last stable circular orbit is at most 3 times this
circr
cmrcmRR circsun1010
2 10~107~
cmkm 61010
cmM
Mkm
M
M
c
GMR
sunsun
52
10332
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Thus in general matter orbits accretor. What happens?
Accretion requires angular momentum loss – see later: specific a.m.at accretor (last orbit) is smaller than initial by factor
Energy loss through dissipation is quicker than angular momentum loss; matter spirals in through a sequence of circular Kepler orbits.
This is an accretion disc. At outer edge a.m. removed by tides fromcompanion star
100)/( 2/1 circrR
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Accretion discs are universal: matter usually has far too much a.m. to accrete directly – matter velocity not `aimed’ precisely at the accretor!
in a galaxy, interstellar gas at radius R from central black holehas specific a.m. , where M is enclosed galaxy mass;far higher than can accrete to the hole, which is
angular momentum increases in dynamical importance as mattergets close to accretor: matter may be captured gravitationally atlarge radius with `low’ a.m. (e.g. from interstellar medium) butstill has far too much a.m. to accrete
Capture rate is an upper limit to the accretion rate
2/1)(~ GMR
cGMcGMGMRGM bhbhbhbhbh /)/(~)(~ 2/122/1
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• expect theory of accretion discs developed below to apply equally to supermassive black—hole accretors in AGN as well as close binaries
• virtually all phenomena present in both cases
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Thin Accretion Discs
Assume disc is closely confined to the orbital plane with semithicknessH, and surface density
in cylindrical polars . Assume also that
These two assumptions are consistent: both require that pressureforces are negligible
Hdz 2
),,( zR 2/1)/( RGMvv K
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Accretion requires angular momentum transport outwards.Mechanism is usually called `viscosity’, but usual `molecular’process is too weak. Need torque G(R) between neighboring annuli
Discuss further later – but functional form must be
with
reason: G(R) must vanish for rigid rotator
Coefficient , where = typical lengthscale andu = typical velocity.
22)( RRRG dRd /
)0(
u ~
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Net torque on disc ring between is
Torque does work at rate
but term
is transport of rotational energy – (a divergence, depending only onboundary conditions).
RRR ,
RR
GRGRRG
)()(
RGGR
RR
G
)(
)(GR
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Remaining term represents dissipation: per unit area (two disc faces!)this is
Note that this is positive, vanishing only for rigid rotation. For Keplerian rotation
and thus
2)(2
1
4)( R
R
GRD
2/13)/( RGM
38
9)(
R
GMRD
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Assume now that disc matter has a small radial velocity .
Then mass conservation requires (exercise!)
Angular momentum conservation is similar, but we must takethe `viscous’ torque into account. The result is (exercise!)
Rv
0)(
RvRRt
R
R
GRvR
RR
tR R
2
1)()( 22
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We can eliminate the radial velocity , and using the Keplerassumption for we get (exercise)
Diffusion equation for surface density: mass diffuses in, angular momentum out.
Diffusion timescale is viscous timescale
Rv
2/12/13
RR
RRRt
/~ 2Rtvisc
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Steady thin discs
Setting we integrate the mass conservation equation as
Clearly constant related to (steady) accretion rate throughdisc as
Angular momentum equation gives
0/ t
constvR R
)(2 RvRM
222 CG
RvR R
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where G(R) is the viscous torque and C a constant.Equation for G(R) gives
Constant C related to rate at which a.m. flows into accretor.If this rotates with angular velocity << Kepler, thereis a point close to the inner edge of the disc where
or equivalently
(sometimes called `no—stress’ boundary condition). Then
32)(
R
CvR
R
0 0)( RG
2/1)(
GMRMC
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Putting this in the equation for and using the Kepler formof angular velocity we get
Using the form of D(R) we find the surface dissipation rate
2/1
13 R
RM
2/1
31
8
3)(
R
R
R
GMMRD
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Now if disc optically thick and radiates roughly as a blackbody,
so effective temperature given by
Note that is independent of viscosity!
is effectively observable, particularly in eclipsing binaries: this confirms simple theory.
4)( bTRD
2/1
3
4 18
3
R
R
R
GMMTb
bT
bT
bT
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Condition for a thin disc (H<<R)
Disc is almost hydrostatic in z-direction, so
But if the disc is thin, z<<R, so this is
2/122 )(
1
zR
GM
zz
P
3
1
R
GMz
z
P
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With
and , where is the sound speed, we find
Hence for a thin disc we require that the local Kepler velocity should be highly supersonic Since this requires that the disc can cool.
HzHPzP ~,/~/ 2~ scP sc
Rv
cR
GM
RcH
K
ss ~~
2/1
If this holds we can also show that the azimuthal velocity isclose to Kepler
2/1Tcs
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thin Keplerian efficiently cooled
Either all three of these properties hold, or none do!
Thus for discs,
*
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u ~u,
scuH ,
Viscosity
Early parametrization: with typical length and velocityscales . Now argue that
First relation obvious, second because supersonic random motionswould shock. Thus set
and argue that . But no reason to suppose
`Alpha—prescription’ useful because disc structure only dependson low powers of . But no predictive power
Hcs
1 const
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Physical angular momentum transport
A disc has
but
accretion requires a mechanism to transport a.m. outwards, but first relation stability against axisymmetric perturbations (Rayleigh criterion).
Most potential mechanisms sensitive to a.m. gradient, so transporta.m. inwards!
,0)( 2
RR
0R
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need a mechanism sensitive to or
Balbus—Hawley (magnetorotational, MRI) instability
magnetic field B threading disc
magnetic tension tries to straighten lineimbalance between gravity and rotation bends line
Kv
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Vertical fieldline perturbed outwards, rotates faster thansurroundings, so centrifugal force > gravity kink increases. Line connects fast-moving (inner) matter with slower (outer) matter, and speeds latter up: outward a.m. transportoutward a.m. transport
if field too strong instability suppressed(shortest growing mode has ) H
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distorted fieldline stretched azimuthally by differential rotation,strength grows
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pressure balance between flux tube and surroundings requires
so gas pressure and hence density lower inside tube buoyant (Parker instability) Flux tube rises
new vertical field, closes cycle
outgasingas PPB
,,
2
8
numerical simulations show this cyclecan transport a.m. efficiently
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Jets
One observed form of outflow: jets with ~ escape velocity from point of ejection, ~ c for black holes
Launching and collimation not understood – probably requirestoroidal magnetic field
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Disc may have two states:
3. infall energy goes into radiation (standard)
5. infall energy goes into winding up internal disc field – thus
disc
generally vertical field directions uncorrelated in neighboringdisc annuli (dynamo random); BUT
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occasionally all fields line up matter swept inwards, strengthensfield energy all goes into field jet ???
(see King, Pringle, West, Livio, 2004)
jets seen (at times) in almost all accreting systems: AGN, LMXBs etc
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Disc timescales
Have met dynamical timescale
and viscous timescale
We define also the thermal timescale
so
2/13 )/(/ GMRvRt Kdyn
/2Rtvisc
viscK
sssth t
R
HR
v
c
GM
cRRDct
2
22
2
2232 )(/
viscthdyn ttt
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Disc stability
Suppose a thin disc has steady—state surface density profile
Investigate stability by setting With so that diffusion equation gives (Exercise)
Thus diffusion (stability) if , butanti—diffusion (instability) if — mass flows towards denser regions, disc breaks up into rings , on viscous timescale.
0
0
)/(
)(3
)( 2/12/1 RR
RRRt
0/
0/
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origin of instability:
so
i.e. local accretion rate increases in response to a decrease in (and vice versa), so local density drops (or rises).
To see condition for onset of instability, recall
M
0/0/
M
4bTM
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and internal temperature T. Thus stability/instabilitydecided by sign of along the equilibrium curvei.e.
C
D
B
A
bT /T
T
0T
0 maxmin
minT
maxT
)(T
)(T
0/ tT
0/ tT
0/ tT
0/ t
A
B
C
D
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Equilibrium
here lies on unstable branch
System is forced to hunt around limit cycle ABCD, unable to reach equilibrium.
evolution A B on long viscous timescaleevolution B C on very short thermal timescaleevolution C D on moderate viscous timescaleevolution C A on very short thermal timescale
Thus get long low states alternating with shorter high states, with rapidupwards and downward transitions between them – dwarf nova light curves.
tT / 0/ t
0/ T
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origin of wiggles in equilibrium curve is hydrogen ionization threshold at
If all of disc is hotter than this, disc remains stably in the highstate – no outbursts.
Thus dwarf novae must have low mass transfer rates:
where is outer disc radius: requires
)(TKT 410~
4163
4 108
3K
R
GMMT
out
b
outR yrMM sun /10~ 10
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Dwarf novae are white dwarf accretors: is there a neutron—star orblack—hole analogue?
soft X—ray transients (SXTs) have outbursts, but much brighter, longerand rarer
why? observation discs are brighter than dwarf novae for same accretion rate X—ray irradiation by central source: disc is concave or warped
(later)
thus not so dominates atlarge R (where most disc mass is)
ionization/stability properties controlled by CENTRAL
2/1 RTT irrb4/3 RTvisc
M
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Thus an SXT outburst cannot be ended by a cooling wave, as in DN.
outburst ends only when central accretion rate drops below acritical value such that
mass of central disc drops significantly long!
KTRT ionoutirr 6500)(
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K & Ritter (1998): in outburst disc is roughly steady—state, with
the central accretion rate. Mass of hot disc is
Now hot zone mass can change only through central accretion, so
3
cM
cM
32
2
0
hc
R
h
RMRdRM
h
ch MM
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thus
i.e.
so central accretion rate, X—rays, drop exponentially for small discs
hh
h MR
M2
3
2/30
hRth eMM
observation indeed shows that short—period (small disc) SXTs areexponential
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condition for SXT outbursts – low disc edge temperature
low mass transfer rate/large disc
observable consequences:
ALL long—period LMXBs are transient outbursts can last years and be separated by many centuriese.g. GRS1915+105: outburst > 15 years
outbursting systems may look persistent
quiescent systems not detectable
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`paradox’ of bright X—ray sources in old stellar systems
elliptical galaxies have sources with :
this requires accretion rates ,
but galaxy has no stars younger than ,
so no extended stars with masses
this would imply X—ray lifetimes i.e. we observeat a very special epoch!
resolution: sources transient, `duty cycle’
sergLX /1039
yrMM sun /10 7
yr1010~
sunM1
yr710
3,210 d
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missing systems:
long—period LMXBs with neutron—star accretors
evolve into millisecond pulsars with white dwarf companions
far too few of former cf latter transients with*
3,210 d
daysP 20~
daysP 100~
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Hyper-Eddington Accretion: SS433
• 13.1—day binary with huge mass transfer rate
• pair of jets (v = 0.26c) precessing with 162—day period, at angle to binary axis
• seen in H alpha, radio, X—rays
• kinetic luminosity of jets ~ erg/s, but radiative luminosity less, e.g. erg/s
• huge outflow (`stationary H alpha’) at 2000 km/s — this is where hyper—Eddington mass flow goes
• this outflow inflates surrounding nebula (W50) and precessing jets make `ears’
39103610xL
o20
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disc warping (Pringle, 1997, 1998)
gravitational potential of accretor ~ spherically symmetric:nothing special about orbital plane – other planes possible, i.e.disc can warp
radiation warping:
photon scattered from surface perturbation
perturbed disc
non—central force torque —disc warps out of original plane
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Tide of companion star on warped disc explains `superorbital periods’ observed in X—ray binaries.m = 0 tide causes 162—d precession —jets precess with disc?
m = 2 tide causes 6.5—d nodding motion—jets show this, but viscosity damps this far too quickly for itto reach centre of disc
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worse –disc is probably aligned in the middle –jets don’t precess at all!
need to make jets sensitive to outer, warped, disc
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outflow can be sensitive to outer disc plane if from large enough R
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outflow bends jets parallel to axis of outer disc, since far more momentum
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Is this reasonable (1) – angle of warped disc?
X—ray observations show disc partially block’s observer’s viewof jets at some precession phases –interpreted as thick disc withH/R ~ 0.4.
But
o204.0tan 1
i.e. ‘thick disc’ is actually warped, tilted outer disc, with jets alongits axis, precessing with it
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(2) If outer disc plane varies slightly, jets should conserve velocity component
cos
along outflow axis. Blundell & Bowler (2005) see just this:
`a cosinusoidal anticorrelation of jet velocity and cone angle’
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Where is the outflow launched?
Shakura & Sunyaev (1973): `spherization radius’
ss
Edd
outsp RR
M
MR ,
4
27Schwarzschild radius
Outflow velocity is v ~ 2000 km/s, suggesting
cmRv
c
v
GMR ssp
102
2
2107
2
for 10 Msun black hole
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these are consistent if
yMMMv
cM sunEddEddout /1033000
27
4 42
2
which agrees well with calculations of thermal—timescale mass transfer (K, Taam & Begelman, 2000)
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within accretion rate must drop as ~R, to keep each radiusbelow Eddington rate. This leads (cf Shakura & Sunyaev, 1973) to
spR
)/ln(28
33 inspEdd
R
R
Eddacc RRLRdR
R
RMGML
sp
in
Now , so logarithm is ~ 10.
Thus a 10 Msun black hole can emit erg/s
insp RR 4104010
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Moreover ‘walls’ of outflow are very optically thick (tau ~ 80)so all luminosity escapes in narrow cone
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An observer viewing the system down this axis would inferan isotropic luminosity
erg/s
where b is the collimation factor.
Ultraluminous X—ray sources (ULXs) may be (non—precessing)systems like this: even with only b = 10% collimation they can reachthe luminosities of observed ULXs.
14010 bL
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active galactic nuclei
supermassive BH in the centre of every galaxy
how did this huge mass grow?
cosmological picture:
)1010( 96sunM
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big galaxy swallows small one
merger
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galaxy mergers
two things happen:
1. black holes coalesce: motion of each is slowed by inertia of gravitational `wake’ – dynamical friction. Sink to bottom of potential and orbit each other. GR emission coalescence
2. accretion: disturbed potential gas near nuclei destabilized, a.m. loss accretion: merged black hole grows: radiation AGN
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black hole coalescence
Hawking’s theorem: black hole event horizon area
or
where a.m., , can never decrease
])/([8 2/122242
4
2
GJcMMc
GA
])1(1[ 2/12*
2 aMA 2
* / GMcJa J
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thus can give up angular momentum and still increase area, i.e.release rotational energy – e.g. as gravitational radiation
then mass M decreases! – minimum is (irreducible mass) – start from and spin down to keeping A fixed
coalescence can be both prograde and retrograde wrt spin of merged hole, i.e. orbital opposite to spin a.m.
Hughes & Blandford (2003): long—term effect of coalescencesis spindown since last stable circular orbit has larger a.m. in retrograde case.
2/M1* a 0* a
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Soltan (1982): total rest—mass energy of all SMBH
consistent with radiation energy of Universe
if masses grew by luminous accretion (efficiency ~10 %)
black hole accretion
thus ADAFs etc unimportant in growing most massiveblack holes
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merger picture of AGN: consequences for accretion
• mergers do not know about black hole mass M, so accretion may be super—Eddington
• mergers do not know about hole spin a, so accretion may be retrograde
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• super—Eddington accretion: must have been common as most SMBH grew (z ~2), so
outflows
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outflow is optically thick to scattering: radiation field L » LEdd transfers » all its momentum to it
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• response to super—Eddington accretion: expel excess accretion as an outflow with thrust given purely by LEdd , i.e.
• outflows with Eddington thrust must have been common as SMBH grew
• NB mechanical energy flux requires knowledge of v or
c
LvM Edd
out
outM c
vLvM Edd
out 2
2
1
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• effect on host galaxy large: must absorb most of the outflow momentum and energy – galaxies not `optically thin’ to matter – unlike radiation
• e.g. could have accreted at » 1Mˉ yr-1 for » 5£107 yr
• mechanical energy deposited in this time » 1060 erg • cf binding energy » 1059 erg of galactic bulge with M » 1011 Mˉ and velocity dispersion » 300 km s-1
• examine effect of super—Eddington accretion on growing SMBH (K 2003)
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• model protogalaxy as an isothermal sphere of dark matter: gas density is
with fg = baryon/matter ' 0.16
• so gas mass inside radius R is
2
2
2)(
Gr
fR g
G
RfdrrRM gR
2
0
2 24)(
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• dynamics depend on whether gas cools (`momentum—driven’) or not (`energy—driven’)
• Compton cooling is efficient out to radius Rc such that M(Rc) » 2£ 10113
200M8
1/2Mˉ
where 200
= /200 km s-1, M8 = M/108Mˉ
• flow is momentum—driven (i.e. gas pressure is unimportant) out to R = Rc
for flow speeds up because of pressure drivingcRR
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swept-up gas
ambient gas outflow
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ram pressure of outflow drives expansion of swept-up shell:
(using M(R) = 2fg2 R/G etc)
thus
constG
fc
L
R
RGMvMvRRRM
dt
d
gEdd
out
4
2
222
4
)(4])([
2000
222
2 222
RtvRtcf
GLR
g
Edd
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for small (i.e. small M), R reaches a maximum
2022
20
202
max 2/2R
cfGL
vRR
gEdd
in a dynamical time
R cannot grow beyond until M grows: expelled matter is trapped inside bubble
M and R grow on Salpeter timescale ~
maxR
yr7105
/~ maxR
EddL
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gas in shell recycled – star formation, chemical enrichment• starbursts accompany black—hole growth
• AGN accrete gas of high metallicity
ultimately shell too large to cool: drives off gas outside
• velocity large: superwind
• remaining gas makes bulge stars — black—hole bulge mass relation
• no fuel for BH after this, so M fixed: M—sigma relation
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42
G
fM g
thus M grows until
or
MM 4200
8102
for a dispersion of 200 km/s
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42
G
fM g
Note: predicted relation
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42
G
fM g has no free parameter!
Note: predicted relation
*
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• M—sigma is very close to observed relation (Ferrarese & Merritt, 2000; Gebhardt et al., 2000; Tremaine et al, 2002)
• only mass inside cooling radius ends as bulge stars, giving
• cooling radius is upper limit to galaxy size
• above results in good agreement with observation
kpcMRc2/1
820080
bulMMM 4/18
4107~
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• argument via Soltan assumes standard accretion efficiency
• but mergersmergers imply accretion flows initially counteraligned in half of all cases, i.e. low accretion efficiency, initial spindown
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• how does SMBH react? i.e. what are torques on hole?
• two main types: 1. accretion – spinup or spindown – requires hole to accrete ~ its own mass to change a/M significantly — slow 2. Lense—Thirring from misaligned disc viscous timescale — fast in inner disc
• standard argument: alignment via Lense—Thirring occurs rapidly, hole spins up to keep a ~ M, accretion efficiency is high
• but L—T also vanishes for counteralignment
• alignment or not? (King, Lubow, Ogilvie & Pringle 05)
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Lense—Thirring:plane of circular geodesic precessesabout black hole spin axis: dissipation causes alignment or counteralignment
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Torque on hole is pure precession, so orthogonal to spin.
Thus general equation for spin evolution is
Here K1, K2 > 0 depend on disc properties. First term specifies precession, second alignment.
Clearly magnitude Jh is constant, and vector sum Jt of Jh, Jd is constant. Thus Jt stays fixed, while tip of Jh moves on a sphere during alignment.
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Using these, we have
thus
But Jh, Jt are constant, so angle h between them obeys
— hole spin always aligns with total angular momentum
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Can further show that Jd2 always decreases during this process –
dissipation
Thus viewed in frame precessing with Jh, Jd,
Jt stays fixed: Jh aligns with it while keeping its length constant
Jd2 decreases monotonically because of dissipation
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Since
there are two cases, depending on whether
or not. If this condition fails, Jt > Jh and alignment follows in the usual way – older treatments implicitly assume
so predicted alignmenthd JJ
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Jh =
Jd =
Jt = Jh + Jd =
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but if does hold,
counteralignment occurs
which requires > and Jd < 2Jh,
then Jt < Jh, and
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• small counterrotating discs anti—align
• large ones align
•what happens in general?
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consider an initially counteraligned accretion event (Lodato & Pringle, 05)
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L—T effect first acts on inner disc: less a.m. than hole, so central disc counteraligns, connected to outer disc by warp: timescale yr810
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but outer disc has more a.m. than hole, so forces it to align, taking counteraligned inner disc with it
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resulting collision of counterrotating gas intense dissipation high central accretion rate
accretion efficiency initially low (retrograde): a/M may be lowertoo
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• merger origin of AGN super—Eddington accretion outflows
• these can explain 1. M—sigma 2. starbursts simultaneous with BH growth 3. BH—bulge mass correlation 4. matter accreting in AGN has high metallicity 5. superwind connection
• about one—half of merger events lead to
1. initial retrograde accretion — low efficiency, lower a/M 2. outbursts