Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 8

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/aigdppt

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Axisymmetric, steady flows
Basic properties
Consequences - an energy conservation
law - an angular momentum law
Application Accretion Disk models
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Basic Definitions
Coordinate system cylindrical polars
Distance element
Gradient operator
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Acceleration in cylindrical polars
Differentiating the components of the velocity
vector
Differentiating the unit vectors
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Cylindrical Polars
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Cylindrical Polars
Curvature term in fluid acceleration
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Fluid acceleration in cylindrical polars
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Axisymmetric flows
Write velocity as polar azimuthal components
Axisymmetry
When acting on scalars
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Fluid accelerationforaxisymmetricflows
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Axisymmetric, steady and frictionless flow
conservation laws
As in any steady frictionless flow Bernoullis
law!
Energy per unit mass
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Angular momentum conservation
Axisymmetric flow, ?-component equation of motion
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Angular momentum conservation
Axisymmetric flow, ?-component equation of motion
Multiply with R and combine blue terms
This is an angular momentum/unit mass specific
angular momentum
This is a torque!
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Angular momentum conservation
Axisymmetric flow, ?-component equation of motion
Use axisymmetry
Conservation of specific angular momentum
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Finally axisymmetric, steady, frictionless flow
Comoving time-derivative
Bernoullis Law Conservation of spec. angular
momentum
Both ? and ? are conserved along flow lines
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Analogy with particle dynamics in central force
field
Effective potential
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The Kepler problem, reduced to one dimension
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Effective potential/energy per unit mass
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Black Hole Kepler Problem
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Conclusion

• Axisymmetry implies conservation of specific
• angular momentum, just like single particle
  dynamics
• In a steady flow, Bernoullis law involves an
  effective
• potential with the same physical
  interpretation as
• in single-particle dynamics.

Immediate consequence
Just like in the Kepler problem, a frictionless
fluid with angular momentum can NOT accrete onto
a compact object due to the Centrifugal Barrier!
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Accretion Disks, where do we find them?
Accreting Binaries
AGNs
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The Eddington Luminosity
Photon momentum flux
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Force on single electron
Outward radial force density
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Force on single electron
Outward radial force density
Eddington Luminosity maximum possible luminosity
for an accreting source where the outward
radiation force balances the inward
gravitational force on the plasma
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Roche-Lobe Overflow
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Geometrically thin accretion disks
Basic assumption thickness ? ltlt radial extent
R Consequences - Disk rotates
quasi-Keplerian - Supported by pressure in
the vertical
direction -
Supported by rotation in the
radial direction - Disk structure
(temperature etc.) set by mass-flow rate
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Basic Geometry definitions
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Thin Disk Approximation
Velocity is in disk plane!
Gravitational potential can be approximated!
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Equations of motion in theThin Disk
Approximation
Effect of viscous friction
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Gravity components
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Force balance in vertical (Z-)directionHydrosta
tic Equilibrium
Basic assumption temperature varies only with
radius
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Force Balance in the radial direction
Order-of-magnitude estimates
Centrifugal term
Pressure gradient
Inertia term
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Inertia term
Pressure gradient
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Conclusion balance between gravity and
centrifugal force in R-direction!
Disk rotates with Keplerian circular angular
velocity!
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Summary Force Balance
Vertical direction Hydrostatic Equilibrium
Radial direction Rotational support
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Mass transport and angular momentum loss
Without friction the specific angular momentum is
conserved!
This implies No Accretion!
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Mass transport and angular momentum loss
Without friction the specific angular momentum is
conserved!
An accretion disk must have friction to get rid
of the angular momentum
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Mass conservation
Continuity equation
Integrate vertically Across disk
? is the surface density disk (in g/cm2)
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Physical interpretation
For a steady flow no accumulation of mass in the
ring
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Angular momentum loss and the radial inflow
velocity
?-component Equation of motion
Radial force balance
Specific angular momentum is also Keplerian
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Height-integrated angular momentum eqn
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Height-integrated angular momentum eqn
Rotation is Keplerian!
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Height-integrated angular momentum eqn
Rotation is Keplerian!
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Conclusion mass flow isproportional toangular
momentum loss
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What determines what the leaky bucket analogy
Faucet Donor star
Water level disk angular
momentum loss
In steady state Mass in per second mass out
per second
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What determines what the leaky bucket analogy
In steady state Mass in per second mass out
per second
Conclusion The water level adjusts until a
steady state is reached with a mass flow set by
the faucet
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What determines what the leaky bucket analogy
In steady state Mass in per second mass out
per second
Conclusion An accretion disk adjusts its
structure until a steady mass flow is reached at
a rate set by the donor star!
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Viscous angular momentum transportapply the
diffusion prescription!

• Angular momentum is transported just like
• ordinary (linear) momentum
• The angular momentum flux follows from the
• diffusion formula in the COROTATING FRAME

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Specific angular momentum in corotating frame
Velocity transformation law
Now apply this in the immediate vicinity of R0
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Specific angular momentum in corotating frame
Velocity difference seen by a corotating
observer for ?RltltR0
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Specific angular momentum in corotating frame
Angular momentum difference seen by a
corotating observer for ?RltltR0
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Diffusive angular momentum flux
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Fraction of particles crossing with angle in
range i, idi
Angular momentum of thermal motion
Flow angular momentum
Radial thermal velocity component
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Average over all inclination angles
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Viscous Torque
For linear momentum force density divergence
viscous
momentum flux
For angular momentum torque divergence
viscous
angular momentum flux
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Viscous Torque
Shear stress
Viscosity coefficient
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Finally the angular momentum equation
Integrate over thickness disk
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Finally the angular momentum equation
Use mass conservation
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Boundary condition at inner disk edge
Matter falls on star Shear stress vanishes!
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Boundary condition at inner disk edge
Disk is Keplerian!
This equation determines disk internal structure!
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Summary Thin Disk equations
Vertical force balance
Radial force balance
Constant mass flux
Relation between mass flux, viscosity and surface
density
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Radial flow speed and theReynolds number
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Radial flow speed and theReynolds number
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From measured disk temperature and accretion rate
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From measured disk temperature and accretion rate
Oops!