OneDimensional Accretion Disk Model with Inflow

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One-Dimensional Accretion Disk Model with Inflow
Outflow

• By
• Peter Becker Truong Le
• Aug. 21, 2001
• George Mason University
• School of Computational Science

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Accretion Disk Background

• The thin accretion disk models describes a
  cooling-dominated flow in which the viscous
  heating of the gas is balanced by local radiative
  cooling.
• Thin accretion disks model first developed by
  Shakura Sunyaev (1973), Novikov Thorne (1973)
• Global models of thin accretion disk developed by
  Paczynski Bisnovatyi-Kogan (1981), Muchotrzeb
  Paczynski (1992)

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Accretion Disk Background

• The thin accretion disk models describes an
  advection-dominated flow in which the radiative
  cooling is very inefficient and most of the
  dissipated energy is advected into the black
  hole.
• Global models Advection-Dominated Accretion
  Flows developed by Begelman (1978), Begelman
  Meier (1982) and Narayan, Kato Honma (1997).

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Accretion Disk Background

• The thin accretion disk models describes an
  advection-dominated flow in which the radiative
  cooling is very inefficient and most of the
  dissipated energy is advected into the black hole
  and some escape to infinity.
• Global models Advection-Dominated Inflow-Outflow
  Solution (ADIOS) developed by Blandford
  Begelman (1999).
• Global models Relativistic Advection-Dominated
  Inflow-Outflow Solutions (RADIOS) modified by
  Becker, Subrammanian Kazanas (2001)

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Accretion Disk Descriptions

• Suppose matter is going around a mass M in a
  nearly circular orbit of radius r.
• Ballancing gravitational force against
  centrifugal force, we find the angular velocity
  to be

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Accretion Disk Descriptions

• Now consider a gaseous disk varies in accordance
  with the angular description W.
• Such a variation of angular velocity would imply
  the existence of velocity shear within the disk
• Due to the action of viscosity, we then expect
  angular momentum to be transferred from the
  faster-moving inner regions of the disk to the
  slower-moving outer regions.
• As the material in an inner layer loses angular
  momentum, it moves inward in a spiral path.

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Accretion Disk Descriptions

• Hence it is viscosity which determines the rate
  of radial inflow of matter and therefore the rate
  at which the gravitational potential energy is
  converted into other forms.
• If the gas had no viscosity, then the material in
  the disk would keep on going in circular orbits
  and there would be no release of gravitational
  energy after the formation of the disk.

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Accretion Disk Flows

• Since the gas will be radially accelerated toward
  the black hole, undergoing a transition from
  subsonic to supersonic speed, a full description
  of the flow necessitates that the pressure and
  inertial terms be included in the radial equation
  motion, the viscosity be included in the angular
  equation of motion, and the energy transport by
  advection be included in the energy equation.
• Also, because we are interested in the outflow
  solution as well as the inflow solution, we need
  to include the outflow of mass, angular momentum
  and energy in the equations of motion.

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4 Conservation Equations

• Mass Conservation
• Angular Momentum Conservation

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4 Conservation Equations

• Radial Momentum Conservation

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4 Conservation Equations

• Energy Conservation

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Narayan et al. (1997) Solution

• To obtain Narayan et al.(1997) solution, we
  assume that there is no outflow by taking the
  escaping times to infinity.
• Once we did that our equations confirm their
  equations of motion.
• At this point, we have confirmed Narayan et
  al.(1997) numerical results by solving these 4
  equations in a time steady state for (v,P,W,r).

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Inner/Sonic Point/Outer Boundary Conditions

• Inner Boundary Conditions
• Entropy parameter (K)
• Specific angular momentum (l)

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Inner Point Boundary Conditions

• Energy flow per unit mass

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Sonic Point Boundary Conditions

• Solving the differential velocity dv/dr we
  obtain two critical point conditions when the
  vanishing of both N and D occurs at the sonic
  point

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Outer Point Boundary Conditions

• Narayan et al.(1997) outer boundary conditions

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Numerical Solution

• Case 1 a.1, gg1.5, J2.6, E-1.01103x10-7,
  K00.00734, rC6.132, rgC2.001

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Numerical Solution

• Case 2 a.001, gg1.5, J3.767 ,
  E-7.03703x10-7, K00.0000708, rC4.22, rgC2.001

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Conclusion

• First, we have shown that our numerical solutions
  are the same as Narayan et al. (1997)
• Second, the Advection-Dominated Accretion Flows
  (ADAF) model provides the picture of how gases
  flow near the vicinity of a black hole
• Third, now that we have understood some what
  about the ADAF inflow model, we could now
  incorporate the outflow part to the model and
  hope to understand the outflow characteristic of
  many AGN objects which contain jet.