Accretion in Binaries Accretion in Binaries –– Basic Basic Physical ProcessesPhysical Processes

Bernhard Müller

12.11.2004

With what meditations did Bloom accompany his demonstrations to his companion of various constellations?[...]of moribund and of nascent stars such as Nova in 1901

J.Joyce, Ulysses

Basics: Energy OutputBasics: Energy Output

Maximum luminosity determined by gravitational potential at the star‘s surface or at the last stable orbit and by the accretion rate:

Typical values for the eff iciency– black hole: 0.057...0.42– neutron star: ≈0.1– white dwarf: ≈10−4

accretionstar

star MR

GML =max

:2cM

L

accretion

=ε

Basics: Eddington LimitBasics: Eddington Limit

z Stable accretion requires that radiation pressure should not outweigh gravitational attraction.

⇒Eddington limit on the luminosity:

⇒Limit on the accretion rate:

sunsunT

pEddington L

M

McGmL

≈

σπ

= 5.4104

m10105.1 418 RaMM sun−−⋅=

When can mass transfer occur When can mass transfer occur in binaries?in binaries?

z Roche model:– Gravitational potential generated by two point

masses

– Transition to corotating frame

– Equations of motions in the new frame (r=distance to CM, r1=distance to star 1, etc.):

22

2

1

1

1

2

1

12

rmr

Gm

r

Gm

P

Ω−−−=Φ

Φ∇−∇ρ

−=×+ u

u

Coriolis term

centrifugal term

Roche potential in the orbital Roche potential in the orbital planeplane

Further assumption: no rotation (in corotating frame)

⇒Equation of hydrostatic equilibrium:

⇒ Star surface (P=const.) will lie on an equipotential surface

Φ∇=∇ρ

P1

Roche potential in the orbital plane

L1

L2

L3

L4

L5

equipotential surfaces of the Roche potential

Types of BinariesTypes of Binaries

http://www.shef.ac.uk/physics/people/vdhillon/seminars/sas/masstrans.html

Stability of Mass TransferStability of Mass Transfer

Roche radius

stellar radii as a function of mass in certain evolutionary phase

mass transfer on thermal timescale or

faster

mass transfer on nuclear timescale

The unfavourable results of fast accretion

expansion within the Roche lobe

Variation of Roche radius with stellar mass

adapted from Padmanabhan, Theroetical Astrophyics

Bondi AccretionBondi Accretionz Assumptions:

– Stationary spherical accretion– Heat losses negligible (for dynamics)– Infalling gas obeys P=KρΓ

z Govering equations:– Continuity equation:

– Bernoulli equation (a=speed of sound):Mur ==ρπ const.4 2

222

1

1const.

1

1

2

1∞−Γ

==−−Γ

+ ar

GMau

P/ρ

Solutions for Spherical Solutions for Spherical AccretionAccretion

z Solutions:– Accretion onto a black hole allows only for

monotonically increasing infall speed⇒unique transonic solution

– Subsonic accretion is possible for neutron stars and white dwarf

– Maximum accretion rate (transonic case):

( )( ) 3

224

∞∞ρΓπλ=

a

cGMM s

Bondi AccretionBondi Accretion

2:radiusaccretion

∞

=a

GMra

http://cfa-www.harvard.edu/~scranmer/News2004/

Bondi Accretion in BinariesBondi Accretion in Binaries

Accretion from (supersonics) stellar winds→completely aspherical!

Heating of gas in the shockwave:

2relvT

m

k ≈∆

Shapiro/Teukolsky

Modified Accretion RateModified Accretion Rate

Gas that has passed the shockfront will be accreted if Ethermal<Egrav:

⇒ Accretion inside a cylinder of radius:

⇒ Modified accretion rate:

22∞+< av

r

GMrel

a

22∞+

≈av

GMr

rela

( ) ( ) 2322

22~

4∞

∞+

ρλπ=av

cGMM

rel

Shapiro/Teukolsky

Luminosity Resulting from Luminosity Resulting from Bondi AccretionBondi Accretion

Main contribution: thermal Bremsstrahlung from the innermost regions (hard X- and γ-rays up to 10MeV)

Luminosity formula:

Bondi accretion is an inefficient mechanism for generating energy:

1

33

4321 serg

10cm1102.1 −

−∞

−∞

⋅≈

sunM

M

K

TnL

⋅≈=ε

−∞

−∞−

sunM

M

K

Tn

cM

L23

4311

2 10cm1106

Disk AccretionDisk Accretion

Roche lobe overflow: accreted material possesses significant angular momentum

⇒ Formation of an accretion disk

Geometrically thin disks require: kTiGMmp/r

⇒ Heat (generated due to disk viscosity) must be radiated away efficiently

Shapiro/Teukolsky

Thickness of a Thin DiskThickness of a Thin Disk

Bernoulli equation for vertical structure:

Integrate:

3

20

2r

GMhP =ρ

∫∫ρ=−

hh

dzr

z

r

GMdz

dz

dP

02

0

r

z

r

GM

dz

dP2

ρ−=

Ω≈ a

h

FG

gradZ P

h

r

Keplerian DisksKeplerian Disks

Fluid moves in Keplerian orbits⇒ Specific angular momentum

j=√GMr⇒ Required outflow of angular

momentum:

which must be achieved by visocous torques:

where

GMrMJ

=

Shapiro/Teukolsky

( )IGMrGMrM

rrhf

−

=⋅π⋅⋅ϕ

22

Ωη=η= ϕϕ

dr

dvf

Nature of Disk ViscosityNature of Disk Viscosity

„Standard“ fluid viscosity due to molecular scattering is far too weak to provide the needed torques and energy dissipation rates.

Alternatives include:– Turbulent viscosity (η=ρvturblturb)

– MHD effects: tangled magnetic fields

Phenomenological modelli ng:

(typical values for α lie between 0.01 and 1)

Resulting luminosity:

1where; ≤αα=φ Pf

R

MGML

β−=

2

3

Structure of the Accretion DiskStructure of the Accretion Disk

r−3/8scatteringradiationinner disk

r−9/10scatteringgasmiddle disk

r−3/4free-free-absorption

gasouter disk

T-dependence

opacity determined

by

dominating P-component

Padmanabhan, Theroetical Astrophyics

Spectrum of an Accreting DiskSpectrum of an Accreting Disk

Emission from an optically thick disk ⇒emission will be locally Planckian

But: modifications for the inner regions

– Opacity dominated by scattering

– Spectrum no longer Planckian!

Further complications: vertical temperature profile,...

( )∫ π⋅−

ννν drr

eI

rkTh 21

~3

Shapiro/Teukolsky

no scattering: absorption after d=1/κabs

scattering: absorption at a distance of d*≈d√ (κabs /κsc)

d d*

random walk

Accretion onto Magnetic Neutron Accretion onto Magnetic Neutron Stars Stars –– Simplified ModelsSimplified Models

Magnetic effects begin to dominate when the magnetic field energy density and the kinetic energy density are comparable:

Alfvén radius for dipole fields and spherical accretion:

Plasma follows the magnetic lines of force inside the Alfven radius

22

8v

B ρ≈π

71

2

4

2

µ=MMG

rA

http://www.ukaff.ac.uk/movies.shtml:

von http://astrosun2.astro.cornell.edu/us-rus/ions.htm

shock front

accretion onto the magnetic poles

Some Other Issues in Some Other Issues in Accretion PhysicsAccretion Physics

More complicated disk structures

Luminosities and spectra for accretion in magnetic fields (synchrotron radiation)

Pulsar spin-up due to magnetic torques

Shocks

Disk instabilities (dwarf novae)

Shapiro/Teukolsky

ReferencesReferences

S.L.Shapiro/S.A.Teukolsky, Black Holes, White Dwarfs and Neutron Stars

T.Padmanabhan, Theoretical Astrophysics vol. 2 (Cambridge Univesity Press)

H.Bondi, On Spherically Symmetrical Accretion (from http://adsabs.harvard.edu/cgi-bin/bib_query?1952MNRAS.112..195B)

http://www.ukaff.ac.uk/movies.shtmlhttp://www.shef.ac.uk/physics/people/vdhillon/seminars/sas/

masstrans.htmlhttp://star-www.st-and.ac.uk/~pja3/movies.html/http://astrosun2.astro.cornell.edu/us-rus/ions.htm