th
8
lecture of “Compact
Object and Accretion”,
Master Programme at Leiden
Observatory
nd
2
Accretion
class
study material: Chapter 4.5, 4.6, 4.7, 4.8, 5,
``accretion power in astrophysics”
these slides at
http://home.strw.leidenuniv.nl/~emr/COA/
Sunday, November 8, 2015
3rd exercise
Quasars & accretion
Articles to read:
• Matthews, T.A. & Sandage, A. R., 1963, ApJ, 138, 30
•Lynden-Bell, 1969, Nature, vol 223
(details are not
important, try to extract may claims, ingredient of
model)
• Pringle, J.E. 1981, ARA&A, 19, 137 (up to section 4
included)
Sunday, November 8, 2015
•
Figure to do:
Plot the flux as a function of frequency (spectrum) of the
radiation coming from an accretion disc for
•
A white dwarf with M=1 Msun, R*= 109 cm, accretion rate
= 10-10 Msun/yr, Rout = 1000 Rin
•
A NS with M=1 Msun, R*= 106 cm, accretion rate = 10-9
Msun/yr, Rout = 100 Rin
•
An AGN with M=108 Msun, R*= 3 Rs , accretion rate = 1
Msun/yr, Rout = 105 Rin
Put then at same distance and assume a line of sight angle i=0
Show on the plot, which part of the spectrum is due to the inner
radius and which to the outer radius. Note the different range of
frequencies covered by the 3 spectra
Sunday, November 8, 2015
Summary content
•
•
State the importance of the discovery of quasars
•
which proposed models could not in the end
explain quasar emission
•
what is the “correct” model and give a simple
energetic argument to support your claim
•
Explain in more detail the correct model, and in
particular how potential energy is extracted
•
End with your own comments/remarks
Sunday, November 8, 2015
What are the main observational properties of
quasars
Accretion discs
First, a simple summary
of main facts/features
Sunday, November 8, 2015
Accretion discs
Both following Roche Lobe overflow, in a semidetached system with stable mass transfer, controlled
by angular momentum loss AND mass transfer in
detached systems via stellar wind, the gas captured
within the Roche Lobe of the compact object is
forced to circularise, via loss of orbital energy
following (self) collisions.
In the case of isolated Supermassive black holes, mass
fuelled from larger distances in their vicinity,
circularizes and forms a disc with size ~0.1-1 pc,
(Lynden-Bell 1969 first to propose disc in quasars)
We now study the physics of the structure (discs) that
transport mass to the compact object: the key
ingredient is the presence of (anomalous) viscosity
Sunday, November 8, 2015
geometrically Thin discs
• 1973-74 Shakura & Sunyaev and independently
Lynden-Bell & Pringle indicate the foundation of thin
discs
•
Same year, relativistic treatment was given by Page
& Thorne
The thin disc approximation implies considering
a disc with hight H << R, at any R. This is possible
when potential energy is efficiently removed as the
gas spirals in towards M1. This occurs for accretion
rates typically below ~0.1 Medd (Medd =Ledd/η c2)
but above ~10-4 Medd
Sunday, November 8, 2015
•
•
•
Radial structure
Eqs. given by hydrodynamical equations, in the case of a
viscous fluid (we will see them later). Assume steady state
The orbits are circular with a tangential velocity that is nearly
Keplerian (see later)
Viscous torques redistribute/transport angular momentum
==> material can accrete radially but much more slowly, on a
viscous timescale. For r >>Rin (inner radius of a disc) the radial
velocity is
with
viscous or
drift timescale
kinematic viscosity
with α <1. α-prescription/parametrization
Shakura & Sunyaev
Sunday, November 8, 2015
Vertical structure
hydrostatic equilibrium
since:
the condition
with
gives:
implies
the Keplerian velocity of a thin disc is highly supersonic. This is realised when
cooling is efficient (the disc is kept “cold”)
Sunday, November 8, 2015
back to radial structure
Euler equation:
•given previous result, the pressure 2term can be neglected w.r.t.
Kepler term:
•what about
<< VK /R
~ VR2/R ?
VR is highly subsonic and VR2/R << cs2/R
Mach number
Sunday, November 8, 2015
disc luminosity
A mass ΔM spiral in from infinity. Its orbital energy
close to the surface of the compact object is
Therefore in Steady state a disc releases a luminosity
Note:
•Ldisc is independent of
viscosity
• For a NW & WD, the inner
radius of the disc is close to
CO surface and indeed 50%
of Lacc is liberated in disc
•For a BH is less. For non
rotating BH we estimated 6%
• 1/2 of Lacc still can be
liberated
Sunday, November 8, 2015
reminder: Lacc: energy released in
radial free fall onto CO “surface”
Temperature/spectrum
•
Each ring emits as a black-body at its surface with a
temperature
σT4 4/3πR2~ Ldisc (R)
the exact formula
come from
combining
conservation laws,
see book
Note: no dependence on viscosity!!!
WD: UV emitter
NS, solar BH: X-ray emitter
(R14~3 Rs)
Sunday, November 8, 2015
AGN (supermassive BH) UV
emitter
Let’s dive a bit more into details
into a steady state, geometrically
thin, optically thick disc
see also Pringle (1981)
1981, ARA&A, 19, 137
Note: we use cylindrical polar coordinates (R, φ,z), quantities are averaged over
the φ-component. When investigating the R direction, the quantities along the zaxis are integrated away. Effectively, one treats R and z direction. All this possible
because H/R <<1 and the disc is symmetric in φ
Sunday, November 8, 2015
Mass accretion in steady state
Given a disc surface density Σ(R) =ρ(R) H(R) of an annulus 2πR dR
mass conservation states that
In steady state (no variation with time, just with R)
where
the amount of viscosity determines the accretion rate and thus
the luminosity (more later)
Sunday, November 8, 2015
Angular momentum transfer
The angular momentum per unit length of a ring is
ΣR2Ω
Angular momentum conservation states:
kinematic viscosity (cm2/s]
•No force with no shearing
•G = torque by the outer ring
onto inner ring. Since <0, the
inner ring loses angular
momentum => flux of
angular momentum outwards
Sunday, November 8, 2015
viscous force per unit length,
acting in φ direction
rate of shearing
<0
see section 3.6 and 4.7, accretion power in astrophysics
Angular momentum transfer in
steady state
C is an integration constant linked to the boundary condition at the
“surface “ of the compact object: is the rate at which angular
momentum flows into the compact star.
Sunday, November 8, 2015
Boundary conditions
(cannot exceed the break up velocity)
+ mass conservation
Sunday, November 8, 2015
Angular momentum transfer in
steady state
***
viscosity controls
the mass accretion
Note: classical result! not true in the presence for example of strong
magnetic field that can force the disc to rotate at its own speed
Let’s use this result to justify 3 previous “just mentioned” results
1.Radial velocity as function of R
~as our dimensional analysis
Sunday, November 8, 2015
2. Disc luminosity: ultimately comes from potential energy
converted into heat
• Viscosity convert potential energy into heat. The work done on a ring of width
dR (force times length) is ~ GdR . The rate of work ~GΩdR.
•
This dissipated heat will be radiated through the two surface of the ring, with
area 4 π R dR. So the “cooling rate per unit surface” is
D(R) =GΩdR /(4 π R dR) = 1/2 υΣ RΩ
with the definition of G
Using ***
there is no viscosity anymore in it!
Note this derivation os done more properly in the book (in 4.6 and 5.3)
Sunday, November 8, 2015
2. Disc luminosity:
we found the result mentioned before!
Sunday, November 8, 2015
2.Note:
the total energy radiated in a ring is
(1
Of this a part comes from the released potential energy between R R+dR
(2
The rest i.e. (1 minus (2 is convected from smaller radii
For R > 4/9 R* is positive => convected energy can be the main source
For R < 4/9 R* is negative => less then released potential energy is irradiated !
Sunday, November 8, 2015
3. Disc temperature as a function of R:
heating
(one side only of the disc)
At equilibrium
Sunday, November 8, 2015
cooling as BB
The spectrum
Each ring emit a BB spectrum with intensity:
Total flux
D = distance, Rout =end of disc , Rin = inner boundary disc
i = line of sight inclination.
The solid angle subtended by a ring is
Sunday, November 8, 2015
The spectrum
υ1/3
υ2
υ3 exp(-hυ/KT)
Tin
Sunday, November 8, 2015
Observations
Sunday, November 8, 2015
•
X-ray binaries
multicolor BB disc emission clearly visible, especially
in the “soft” state, in the X-ray band
Cygnus X1
Sunday, November 8, 2015
CVs
Disc in optical/UV and P-Cygni line profiles in UV, indicating strong
outflows together with accretion
Sunday, November 8, 2015
Active Galactic Nuclei
The disc emission is the “Blue bump” in UVsoft X-rays? see book in section 8.1
Sunday, November 8, 2015