Effects of Compressibility, Turbulent Viscosity and Mass-Transfer-Rate on

Accretion Discs in Close Binaries: Timescales of Outburst Events

G. Lanzafame1, V. Costa2, G. Belvedere3

1 INAF-Osservatorio Astrofisico di Catania

2 Dip. M.F.C.l. - Università di Catania

3 Dip Fisica e Astronomia – Università di Catania

• In accretion discs in close binaries• Study of the variability of the emitted power through the disc’s

variability of the its hydrodynamic structure;• Application of the SPH numerical technique (Lagrangian numerical

scheme);• SPH origin to be found in Lucy (1977) + Monaghan (from 1977 up to

today - in particular 1992);• PM methods vs. SPH: in SPH, spatial interpolations are performed

on particles themselves;• SPH widely adopted also in laboratory flow simulations or in solid-

body collision simulations (Astrophysical and Engineering);• SPH provides better results than other methods in simulations of

flow discontinuities and/or convective fows;

133136 10;10 sergLsergL optx

• In SPH, (interpolation or convolution integral)

• Kernel – smooth function: cubic spline or Gaussian function;

• Normalization:

• In computational discretization,

• , = SPH particle mass.

• For viscous ideal gas dynamics:

• Define the viscous stress tensor component

• For viscous flows, set as the specific total energy

jijjj jjijj ji WAmnWAA //

hrWhrrWW ijjiij ,,,

rdWAA ijji3

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2

1

• ideal gas state equation (1)

• continuity equation (2)

• momentum equation (3)

• energy equation (4)

• In SPH framework, all spatial derivatives of physical quantities transform in Kernel spatial derivatives. SPH particle mass is known. Therefore SPH particle density can be computed by definition as:

• (SPH equiv. of the time integral of continuity eq. 2)

• Or via temporal integration of direct SPH conversion of eq. (2):

i

ii

i

ii p

dt

vd

iii v

dt

d

iii

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ijj ji Wm

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• continuity equation (5)

•

• As for momentum equation and energy equation, we consider only the shear contribution of the viscous stress tensor and the relative first viscosity coefficient , excluding the bulk viscosity contribution for the sake of simplicity, and parametrize adopting the Shakura and Sunyaev parametrization:

• , , ,

• Such a turbulent physical viscosity is included only in viscous modelling. Instead artificial viscosity is necessary in inviscid SPH fluid dynamics;

• Therefore, the SPH conversion of momentum equation and energy equation in viscous modelling is:

1

ijiijj ji Wvm

dt

d

jiij vvv

vv Hcsv

v

211 xysxy rMcrH

• momentum equation (6)

•

• energy equation (7)

•

iijij

j

i

ij j

i Wpp

mdt

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ijij

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ijij

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Applications: Accretion Discs in Close Binaries

• , Km in our models

• Non-dimensional equations - Normalization factors:

• hours (SU Uma, OY Car and Z Cha-like)

MMM 1, 21 612 10r

121 MMM

112 rr

1orbit

5orbP

Boundary conditions at disc inner and outer edges

• Disc inner edge:

Particle free fall onto the primary star. Particle are simply eliminated when their radial distance from the primary star

• Disc outer edge:

Particle lost in the outer space. Particle are eliminated when their radial distance from the primary star where is the radial distance of the inner Lagrangian point L1.

hrr in 21

11 Lout rrr 1Lr

Injection conditions in L1

• The injection of “new” particles from L1 toward the interior of the primary Roche Lobe is simulated by generating them in fixed points (injectors) symmetrically placed within an angle having L1 as a vertex and a conic aperture of 1 radiant. The initial velocity of injected particles is radial with respect to L1. in order to simulate a regular and smooth gas injection, a “new” particle is generated in the injectors whenever “old” particles are far enough from the injectors with respect to the SPH resolution length (resolving power) .

Stationary state disc models

• Stationary state disc configuration is fulfilled when the total number of disc particles is statistically constant (balance among particles injected, accreted and ejected).

h

Basic results (2003 – 2006)

• The results, obtained adopting , show that turbulent physical viscosity supports and favours accretion disc formation in the low compressibility case ( ) , but no spiral shocks develop.

• Spiral shocks in the radial flux develop in all high compressibility models ( ). Physical viscosity efficiently supports mass, angular momentum and heat radial transport towards the compact primary star as well as the radial disc spread reduction.

• Flebbe et al. 1994, ApJ 431, 754.• Lanzafame 2003, A&A 403, 593. • Lanzafame et al. 2006, A&A 453, 1027.• Shakura and Sunyaev 1973, A&A 24, 337.

1

01.1

3/5

• Fig. 1

Dwarf Novae outbursts

• Currently, a typical turbulent Shakura and Sunyaev parameter is largely adopted. Investigation is here carried out both in high compressibility regimes and in low compressibility ones, with the aim to evaluate, in a compressibility-viscosity graph, the most suitable dominion where physical conditions allows a well-bound disc development as a function of mass transfer kinematic conditions;

• Results show that domains exists, where turbulent physical viscosity supports the accretion disc formation. Usually the lower the gas compressibility (the higher is the polytropic index ) and the higher the physical viscosity requested. A role also played by the injection kinematics at the inner Lagrangian point L1 is also found;

• Conclusions as far as dwarf novae outbursts are concerned;

1.0

• Fig. 2

• Novae outburst simulation:

• Active to quiescent Quiescent to active

• Transition lasts orbital periods; Transition lasts orbital periods

sgM acc /1010 1617

sgM acc /1010 1716

10 5

sKmvinj /13013),1.0,5/7(

• Dwarf novae outburst, as well as quiescent phases, can be related to compressibility-viscosity state in the ( ) diagram, according to the boundary line referring to the kinematical injection velocity at the L1 point. Transitions from one state to another, or from one kinematical bounday line to another are theoretically arbitrary. Therefore a transition from one nova phase to another could be realized varying periodically and/or and/or ;

• Outburst phases are related to well-bound accretion discs in the primary’s potential well, whose ( ) pair is located above the kinematic boundary line defined by the mass transfer kinematics from the L1 point. Quiescent phases are related to rarefied accretion disc structures in the primary’s potential well in low compressibility regimes and to well bound accretion discs in the primary’s potential well in high compressibility regimes, whose ( ) pair is located below the kinematic boundary line defined by the mass transfer kinematics from the L1 point;

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injv

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injv