8 lecture of “compact object and accretion”, master...

8th lecture of “Compact Object and Accretion”, Master Programme at Leiden

Observatory

Accretion 2nd classstudy 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 exerciseQuasars & 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)


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


Summary content• State the importance of the discovery of quasars

• What are the main observational properties 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


First, a simple summary of main facts/features

Accretion discs


Accretion discsBoth following Roche Lobe overflow, in a semi-detached 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


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


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

viscous or drift timescale

with

kinematic viscosity

with α <1. α-prescription/parametrization Shakura & Sunyaev


Vertical structurehydrostatic equilibrium

since: with gives:

the condition implies

the Keplerian velocity of a thin disc is highly supersonic. This is realised when cooling is efficient (the disc is kept “cold”)


<< VK2/R

back to radial structureEuler equation:

•given previous result, the pressure term can be neglected w.r.t. Kepler term:

•what about ~ VR2/R ?

VR is highly subsonic and VR2/R << cs2/R

Mach number


disc luminosityA 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

reminder: Lacc: energy released inradial free fall onto CO “surface”

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


Temperature/spectrum• Each ring emits as a black-body at its surface with a

temperature

WD: UV emitterNS, solar BH: X-ray emitter

σT4 4/3πR2~ Ldisc (R) the exact formula

come from combining

conservation laws, see book

Note: no dependence on viscosity!!!

AGN (supermassive BH) UV emitter(R14~3 Rs)


Let’s dive a bit more into details into a steady state, geometrically

thin, optically thick discsee 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 z-axis are integrated away. Effectively, one treats R and z direction. All this possible

because H/R <<1 and the disc is symmetric in φ


http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1981ARA%26A..19..137P&db_key=AST&link_type=ABSTRACT&high=487a472aff23470

http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1981ARA%26A..19..137P&db_key=AST&link_type=ABSTRACT&high=487a472aff23470

Mass accretion in steady stateGiven 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)


Angular momentum transferThe angular momentum per unit length of a ring is

ΣR2ΩAngular momentum conservation states:

viscous force per unit length,acting in φ direction rate of shearing

see section 3.6 and 4.7, accretion power in astrophysics

kinematic viscosity (cm2/s]

<0

•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


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.


Boundary conditions

(cannot exceed the break up velocity)

+ mass conservation


Angular momentum transfer in steady state

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

viscosity controls the mass accretion

Let’s use this result to justify 3 previous “just mentioned” results

1.Radial velocity as function of R

***

~as our dimensional analysis


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 ***

Note this derivation os done more properly in the book (in 4.6 and 5.3)

there is no viscosity anymore in it!


2. Disc luminosity:

we found the result mentioned before!


2.Note: the total energy radiated in a ring is

Of this a part comes from the released potential energy between R R+dR

(1

(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 !


3. Disc temperature as a function of R:

heating(one side only of the disc) cooling as BB

At equilibrium


The spectrumEach ring emit a BB spectrum with intensity:

Total flux

D = distance, Rout =end of disc , Rin = inner boundary disci = line of sight inclination. The solid angle subtended by a ring is


The spectrum

υ1/3

υ2

υ3 exp(-hυ/KT)

Tin


Observations


X-ray binaries• multicolor BB disc emission clearly visible, especially

in the “soft” state, in the X-ray band

Cygnus X1


CVs

Disc in optical/UV and P-Cygni line profiles in UV, indicating strong outflows together with accretion


Active Galactic NucleiThe disc emission is the “Blue bump” in UV-soft X-rays? see book in section 8.1


8 lecture of “compact object and accretion”, master...

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